{"id":4528,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/"},"modified":"2018-05-08T17:31:55","modified_gmt":"2018-05-08T12:01:55","slug":"ncert-solutions-class-12-maths-miscellaneous-7","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/","title":{"rendered":"NCERT Solutions class 12 Maths Miscellaneous"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Vector_Algebra\" >NCERT Solutions class 12 Maths Vector Algebra<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#2_Find_the_scalar_components_and_magnitude_of_the_vector_joining_the_points_P_and_Q\" >2. Find the scalar components and magnitude of the vector joining the points P\u00a0and Q<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#3_A_girl_walks_4_km_towards_west_then_she_walks_3_km_in_a_direction_east_of_north_and_stops_Determine_the_girls_displacement_from_her_initial_point_of_departure\" >3. A girl walks 4 km towards west, then she walks 3 km in a direction \u00a0east of north and stops. Determine the girl\u2019s displacement from her initial point of departure.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#5_Find_the_value_of_for_which_is_a_unit_vector\" >5. Find the value of \u00a0for which \u00a0is a unit vector.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#6_Find_a_vector_of_magnitude_5_units_and_parallel_to_the_resultant_of_the_vectors_and\" >6. Find a vector of magnitude 5 units and parallel to the resultant of the vectors \u00a0and \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#7_If_and_find_a_unit_vector_parallel_to_the_vector\" >7. If \u00a0and \u00a0find a unit vector parallel to the vector \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#8_Show_that_the_points_A_B_and_C_11_3_7_are_collinear_and_find_the_ratio_in_which_B_divides_AC\" >8. Show that the points A\u00a0B\u00a0and C (11, 3, 7) are collinear and find the ratio in which B divides AC.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#9_Find_the_position_vector_of_a_point_R_which_divides_the_line_joining_the_two_points_P_and_Q_whose_position_vectors_are_and_externally_in_the_ratio_1_2_Also_show_that_P_is_the_middle_point_of_line_segment_RQ\" >9. Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are \u00a0and \u00a0externally in the ratio 1 : 2. Also, show that P is the middle point of line segment RQ.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#10_Two_adjacent_sides_of_a_parallelogram_are_and_Find_the_unit_vector_parallel_to_its_diagonal_Also_find_its_area\" >10. Two adjacent sides of a parallelogram are \u00a0and \u00a0Find the unit vector parallel to its diagonal. Also, find its area.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#11_Show_that_the_direction_cosines_of_a_vector_equally_inclined_to_the_axes_OX_OY_and_OZ_are\" >11. Show that the direction cosines of a vector equally inclined \u00a0to the axes OX, OY and OZ are ,,.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#12_Let_and_Find_a_vector_which_is_perpendicular_to_both_and_and\" >12. Let \u00a0and \u00a0Find a vector \u00a0which is perpendicular to both \u00a0and \u00a0and \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-2\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#13_The_scalar_product_of_the_vector_with_a_unit_vector_along_the_sum_of_vectors_and_is_equal_to_one_Find_the_value_of\" >13. The scalar product of the vector \u00a0with a unit vector along the sum of vectors \u00a0and \u00a0is equal to one. Find the value of \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-3\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#14_If_are_mutually_perpendicular_vectors_of_equal_magnitudes_show_that_the_vector_is_equally_inclined_to\" >14. If \u00a0are mutually perpendicular vectors of equal magnitudes, show that the vector \u00a0is equally inclined to .<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-4\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#15_Prove_that_if_and_only_if_are_perpendicular_given\" >15. Prove that \u00a0if and only if \u00a0are perpendicular given \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-5\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#16_Choose_the_correct_answer\" >16. Choose the correct answer:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-6\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#17_Choose_the_correct_answer\" >17. Choose the correct answer:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-7\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#18_Choose_the_correct_answer\" >18. Choose the correct answer:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-8\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#19_If_be_the_angle_between_any_two_vectors_and_then_when_is_equal_to\" >19. If \u00a0be the angle between any two vectors \u00a0and , then \u00a0when \u00a0is equal to:<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#NCERT_Solutions_class_12_Maths_Miscellaneous-9\" >NCERT Solutions class 12 Maths Miscellaneous<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.<\/p>\n<p><strong>Download\u00a0<a href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics-vector-algebra\/1294\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Vector Algebra\u00a0<\/a>as PDF.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/media-mycbseguide.s3.ap-south-1.amazonaws.com\/images\/blog\/12%20Maths%20Book%20Part-02%20.jpg\" alt=\"NCERT Solutions class 12 Maths Miscellaneous\" width=\"203\" height=\"260\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Vector_Algebra\"><\/span>NCERT Solutions class 12 Maths Vector Algebra<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>1. Write down a unit vector in XY-plane making an angle of <\/strong><img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image001.png\" \/><strong>\u00a0with the positive direction of <\/strong><img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image002.png\" \/><strong>axis.<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 23px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image003.png\" \/>\u00a0be the unit vector in XY-plane such that <img decoding=\"async\" style=\"height: 16px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image004.png\" \/>XOP = <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image005.png\" \/><\/p>\n<p><img decoding=\"async\" id=\"Picture 6\" style=\"height: 204px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image006.jpg\" \/><\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 32px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image007.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0\u00a0OP = 1\u00a0\u00a0 \u2026\u2026\u2026.(i)<\/p>\n<p>By Triangle Law of Addition of vectors,<\/p>\n<p>In <img decoding=\"async\" style=\"height: 19px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image009.png\" \/>, <img decoding=\"async\" style=\"height: 23px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image010.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 29px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image011.png\" \/><\/p>\n[Unit vector along OX is <img decoding=\"async\" style=\"height: 24px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image012.png\" \/>\u00a0and that is along OY is <img decoding=\"async\" style=\"height: 27px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image013.png\" \/>]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image014.png\" \/>\u00a0 [Dividing and multiplying by OP in R.H.S.]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image015.png\" \/>\u00a0[Using eq. (i)]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image016.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"2_Find_the_scalar_components_and_magnitude_of_the_vector_joining_the_points_P_and_Q\"><\/span><strong>2. Find the scalar components and magnitude of the vector joining the points P<\/strong><img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image017.png\" \/><strong>\u00a0and Q<\/strong><img decoding=\"async\" style=\"height: 27px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image018.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given points are P<img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image017.png\" \/> and Q<img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image019.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0Position vector of point P = <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image017.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 28px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image020.png\" \/><\/p>\n<p>And Position vector of point Q = <img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image019.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 28px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image021.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 25px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image022.png\" \/>\u00a0\u00a0= Position vector of Q \u2013 Position vector of P<\/p>\n<p>= <img decoding=\"async\" style=\"height: 28px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image021.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image023.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 28px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image021.png\" \/><img decoding=\"async\" style=\"height: 28px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image024.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 29px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image025.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 Scalar components of the vector <img decoding=\"async\" style=\"height: 25px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image022.png\" \/>\u00a0are the coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image027.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 25px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image022.png\" \/>, i.e.,<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image028.png\" \/><\/p>\n<p>And magnitude of vector <img decoding=\"async\" style=\"height: 25px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image022.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image029.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"3_A_girl_walks_4_km_towards_west_then_she_walks_3_km_in_a_direction_east_of_north_and_stops_Determine_the_girls_displacement_from_her_initial_point_of_departure\"><\/span><strong>3. A girl walks 4 km towards west, then she walks 3 km in a direction <\/strong><img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image001.png\" \/><strong>\u00a0east of north and stops. Determine the girl\u2019s displacement from her initial point of departure.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let the initial point of departure is origin (0, 0) and the girl walks a distance OA = 4 km towards west.<\/p>\n<p>Through the point A, draw a line AQ parallel to a line OP, which is <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image001.png\" \/>\u00a0East of North, i.e., in East-North quadrant making an angle of <img decoding=\"async\" style=\"height: 21px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image001.png\" \/>\u00a0with North.<\/p>\n<p>Again, let the girl walks a distance AB = 3 km along this direction <img decoding=\"async\" style=\"height: 25px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image030.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image031.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image032.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)\u00a0 [<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image033.png\" \/>\u00a0Vector <img decoding=\"async\" style=\"height: 23px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image034.png\" \/>\u00a0is along OX\u2019]\n<p><img decoding=\"async\" id=\"Picture 7\" style=\"height: 177px; width: 239px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image035.jpg\" \/><\/p>\n<p>Now, draw BM perpendicular to <img decoding=\"async\" style=\"height: 15px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image002.png\" \/>axis.<\/p>\n<p>In <img decoding=\"async\" style=\"height: 21px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image036.png\" \/>\u00a0by Triangle Law of Addition of vectors,<\/p>\n<p><img decoding=\"async\" style=\"height: 29px; width: 227px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image037.png\" \/><\/p>\n<p>Dividing and multiplying by AB in R.H.S.,<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 177px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image038.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image039.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image040.png\" \/>\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 Girl\u2019s displacement from her initial point O of departure to final point B,<\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image041.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 53px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image042.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 48px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image043.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image044.png\" \/><\/p>\n<hr \/>\n<p><strong>4. If <\/strong><img decoding=\"async\" style=\"height: 25px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image045.png\" \/><strong>\u00a0then is it true that <\/strong><img decoding=\"async\" style=\"height: 32px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image046.png\" \/><strong>\u00a0Justify your answer.<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image047.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0Either the vectors <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image048.png\" \/>\u00a0are collinear or form the sides of a triangle.<\/p>\n<p>Case I: Vectors <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image048.png\" \/>\u00a0are collinear.<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 25px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image049.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image050.png\" \/><\/p>\n<p>Then <img decoding=\"async\" style=\"height: 23px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image051.png\" \/><\/p>\n<p>Also, <img decoding=\"async\" style=\"height: 32px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image052.png\" \/>\u00a0= AC = AB + BC = <img decoding=\"async\" style=\"height: 32px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image053.png\" \/><\/p>\n<p>Case II: Vectors <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image048.png\" \/>\u00a0form a triangle.<\/p>\n<p>Here also by Triangle Law of vectors, <img decoding=\"async\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image047.png\" \/><\/p>\n<p>But <img decoding=\"async\" style=\"height: 32px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image054.png\" \/>\u00a0\u00a0[<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image033.png\" \/>\u00a0Each side of a triangle is less than sum of the other two sides]\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image055.png\" \/>\u00a0is true only when vectors <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image057.png\" \/>\u00a0are collinear vectors.<\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"5_Find_the_value_of_for_which_is_a_unit_vector\"><\/span><strong>5. Find the value of <\/strong><img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image058.png\" \/><strong>\u00a0for which <\/strong><img decoding=\"async\" style=\"height: 35px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image059.png\" \/><strong>\u00a0is a unit vector.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Since <img decoding=\"async\" style=\"height: 35px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image060.png\" \/>\u00a0is a unit vector,<\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 35px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image061.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image062.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image063.png\" \/><\/p>\n<p>Squaring both sides,\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image064.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image065.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image066.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"6_Find_a_vector_of_magnitude_5_units_and_parallel_to_the_resultant_of_the_vectors_and\"><\/span><strong>6. Find a vector of magnitude 5 units and parallel to the resultant of the vectors <\/strong><img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image067.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image068.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Vectors <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image067.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image069.png\" \/><\/p>\n<p>Let vector <img decoding=\"async\" style=\"height: 23px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image057.png\" \/>\u00a0be the resultant vector of <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image071.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image072.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image073.png\" \/>\u00a0+ <img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image074.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image075.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0Required vector pf magnitude 5 units and parallel (or collinear) to resultant vector <img decoding=\"async\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image072.png\" \/>\u00a0is<\/p>\n<p><img decoding=\"async\" style=\"height: 57px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image076.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 44px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image077.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 48px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image078.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 45px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image079.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 45px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image080.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 45px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image081.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"7_If_and_find_a_unit_vector_parallel_to_the_vector\"><\/span><strong>7. If <\/strong><img decoding=\"async\" style=\"height: 27px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image082.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image083.png\" \/><strong>\u00a0find a unit vector parallel to the vector <\/strong><img decoding=\"async\" style=\"height: 23px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image084.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Vectors <img decoding=\"async\" style=\"height: 27px; width: 173px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image082.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image085.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 23px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image086.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 261px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image087.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 245px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image088.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image089.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 A unit vector parallel to the vector <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image090.png\" \/>\u00a0is<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image091.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 48px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image092.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 44px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image093.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"8_Show_that_the_points_A_B_and_C_11_3_7_are_collinear_and_find_the_ratio_in_which_B_divides_AC\"><\/span><strong>8. Show that the points A<\/strong><img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image094.png\" \/><strong>\u00a0B<\/strong><img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image095.png\" \/><strong>\u00a0and C (11, 3, 7) are collinear and find the ratio in which B divides AC.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Points A<img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image094.png\" \/>\u00a0B<img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image096.png\" \/>\u00a0and C (11, 3, 7).<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 Position vector of point A = <img decoding=\"async\" style=\"height: 29px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image097.png\" \/><\/p>\n<p>Position vector of point B = <img decoding=\"async\" style=\"height: 29px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image098.png\" \/><\/p>\n<p>Position vector of point C = <img decoding=\"async\" style=\"height: 29px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image099.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image100.png\" \/>\u00a0\u00a0= Position vector of point B \u2013 Position vector of point A<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image101.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image102.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image103.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image104.png\" \/><\/p>\n<p>Again <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image105.png\" \/>\u00a0\u00a0= Position vector of point C \u2013 Position vector of point B<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image106.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image107.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image108.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 261px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image109.png\" \/><\/p>\n<p>Again\u00a0 <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image110.png\" \/>\u00a0\u00a0= Position vector of point C \u2013 Position vector of point A<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 175px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image111.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image112.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image113.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 289px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image114.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 27px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image115.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image113.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image110.png\" \/><\/p>\n<p>Therefore, points A, B, C are either collinear or are the vertices of a triangle ABC.<\/p>\n<p>Again AB + BC = <img decoding=\"async\" style=\"height: 23px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image116.png\" \/>\u00a0= AC<\/p>\n<p><strong>Now to find ratio in which B divides AC<\/strong><\/p>\n<p>Let the point B divides AC in the ratio <img decoding=\"async\" style=\"height: 19px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image117.png\" \/><\/p>\n<p>Therefore, using section formula, Position vector of point B<img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image096.png\" \/>\u00a0 is <img decoding=\"async\" style=\"height: 45px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image118.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image096.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image119.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 259px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image120.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 343px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image121.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 341px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image122.png\" \/><\/p>\n<p>Comparing coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image027.png\" \/>\u00a0both sides, we get<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 324px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image123.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 373px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image124.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image125.png\" \/><\/p>\n<p>Therefore, required ratio = <img decoding=\"async\" style=\"height: 19px; width: 29px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image126.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image127.png\" \/>\u00a0: 1 = 2 : 3<\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"9_Find_the_position_vector_of_a_point_R_which_divides_the_line_joining_the_two_points_P_and_Q_whose_position_vectors_are_and_externally_in_the_ratio_1_2_Also_show_that_P_is_the_middle_point_of_line_segment_RQ\"><\/span><strong>9. Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are <\/strong><img decoding=\"async\" style=\"height: 32px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image128.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 32px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image129.png\" \/><strong>\u00a0externally in the ratio 1 : 2. Also, show that P is the middle point of line segment RQ.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Since position vector of point R dividing the join of P and Q externally in the ratio 1 : 2 = <img decoding=\"async\" style=\"height: 15px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image130.png\" \/>\u00a0is given by <img decoding=\"async\" style=\"height: 45px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image131.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 51px; width: 163px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image132.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 45px; width: 263px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image133.png\" \/><\/p>\n<p>Again position vector of the middle point of the line segment RQ<\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image134.png\" \/>\u00a0(Position vector of point R \u2013 Position vector of point Q)<\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image135.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image136.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 23px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image137.png\" \/>\u00a0= Position vector of point P (given)<\/p>\n<p>Therefore, P is the middle point of the line segment RQ.<\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"10_Two_adjacent_sides_of_a_parallelogram_are_and_Find_the_unit_vector_parallel_to_its_diagonal_Also_find_its_area\"><\/span><strong>10. Two adjacent sides of a parallelogram are <\/strong><img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image138.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image139.png\" \/><strong>\u00a0Find the unit vector parallel to its diagonal. Also, find its area.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let ABCD is a parallelogram.<\/p>\n<p><img decoding=\"async\" id=\"Picture 8\" style=\"height: 127px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image140.png\" \/><\/p>\n<p>Given: The vectors representing two adjacent sides of this parallelogram say,<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image141.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image142.png\" \/><\/p>\n<p>Now vectors along the diagonals <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image110.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image143.png\" \/>\u00a0of the parallelogram are<\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image144.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image145.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 23px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image144.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image146.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image147.png\" \/><\/p>\n<p>And <img decoding=\"async\" style=\"height: 23px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image145.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image148.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image149.png\" \/><\/p>\n<p>Therefore, Unit vectors parallel to (or along) diagonals are<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image150.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 56px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image151.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 48px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image152.png\" \/>\u00a0\u00a0and <img decoding=\"async\" style=\"height: 48px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image153.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image154.png\" \/>\u00a0\u00a0and <img decoding=\"async\" style=\"height: 48px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image155.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image156.png\" \/>\u00a0 and <img decoding=\"async\" style=\"height: 48px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image155.png\" \/><\/p>\n<p>Now Area of parallelogram = <img decoding=\"async\" style=\"height: 23px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image157.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 83px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image158.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 29px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image159.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image160.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 36px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image161.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image162.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image163.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image164.png\" \/>\u00a0sq. units<\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"11_Show_that_the_direction_cosines_of_a_vector_equally_inclined_to_the_axes_OX_OY_and_OZ_are\"><\/span><strong>11. Show that the direction cosines of a vector equally inclined <\/strong><img decoding=\"async\" style=\"height: 44px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image165.png\" \/><strong>\u00a0to the axes OX, OY and OZ are <\/strong><img decoding=\"async\" style=\"height: 44px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image165.png\" \/><strong>,<\/strong><img decoding=\"async\" style=\"height: 44px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image165.png\" \/><strong>,<\/strong><img decoding=\"async\" style=\"height: 44px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image165.png\" \/><strong>.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image166.png\" \/>\u00a0be the direction cosines of a vector equally inclined to axes OX, OY and OZ respectively.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 A unit vector along the given vector is<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image167.png\" \/>\u00a0\u00a0and <img decoding=\"async\" style=\"height: 35px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image168.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image169.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image170.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Let the given vector (for which unit vector is <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image171.png\" \/>) make equal angle (given) <img decoding=\"async\" style=\"height: 21px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image172.png\" \/>\u00a0(say) with OX<img decoding=\"async\" style=\"height: 35px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image173.png\" \/>\u00a0OY<img decoding=\"async\" style=\"height: 35px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image174.png\" \/>\u00a0and OZ<img decoding=\"async\" style=\"height: 35px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image175.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0The given vector is in positive octant OXYZ and hence <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0is acute. \u2026\u2026\u2026.(ii)<\/p>\n<p>Now angle <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0between <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image171.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 24px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image012.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 57px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image177.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 57px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image178.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image179.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image180.png\" \/>\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(iii)<\/p>\n<p>Similarly, angle <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0between <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image171.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image013.png\" \/>, <img decoding=\"async\" style=\"height: 19px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image181.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(iv)<\/p>\n<p>And angle <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0between <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image171.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image182.png\" \/>, <img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image183.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(v)<\/p>\n<p>Putting the values of <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image166.png\" \/>\u00a0in eq. (i), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image184.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 21px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image185.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image186.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image187.png\" \/><\/p>\n<p>But <img decoding=\"async\" style=\"height: 44px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image188.png\" \/>\u00a0[<img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image033.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0is acute and hence <img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image189.png\" \/>\u00a0is positive]\n<p>Therefore, required vectors <img decoding=\"async\" style=\"height: 21px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image166.png\" \/>\u00a0are <img decoding=\"async\" style=\"height: 44px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image190.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 44px; width: 31px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image191.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"12_Let_and_Find_a_vector_which_is_perpendicular_to_both_and_and\"><\/span><strong>12. Let <\/strong><img decoding=\"async\" style=\"height: 27px; width: 199px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image192.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image193.png\" \/><strong>\u00a0Find a vector <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/><strong>\u00a0which is perpendicular to both <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image194.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Vectors <img decoding=\"async\" style=\"height: 27px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image195.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image196.png\" \/><\/p>\n<p>We know that the cross-product of two vectors, <img decoding=\"async\" style=\"height: 23px; width: 33px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image157.png\" \/>\u00a0is a vector perpendicular to both <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image071.png\" \/><\/p>\n<p>Hence, vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0which is also perpendicular to both <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 32px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image197.png\" \/>\u00a0where <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image198.png\" \/>\u00a0or some other scalar.<\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 83px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image199.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 235px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image200.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 35px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image201.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026..(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 140px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image202.png\" \/><\/p>\n<p>Now given <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image203.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image204.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image204.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image205.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image206.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image207.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image208.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image209.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 41px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image209.png\" \/>\u00a0in eq. (i), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image210.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image211.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-2\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"13_The_scalar_product_of_the_vector_with_a_unit_vector_along_the_sum_of_vectors_and_is_equal_to_one_Find_the_value_of\"><\/span><strong>13. The scalar product of the vector <\/strong><img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image212.png\" \/><strong>\u00a0with a unit vector along the sum of vectors <\/strong><img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image213.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image214.png\" \/><strong>\u00a0is equal to one. Find the value of <\/strong><img decoding=\"async\" style=\"height: 19px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image215.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image216.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image217.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image218.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 23px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image219.png\" \/>\u00a0(say) = <img decoding=\"async\" style=\"height: 29px; width: 117px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image220.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 24px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image221.png\" \/>\u00a0a unit vector along <img decoding=\"async\" style=\"height: 23px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image219.png\" \/>\u00a0is<\/p>\n<p><img decoding=\"async\" style=\"height: 24px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image221.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 59px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image222.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 51px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image223.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 51px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image224.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 24px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image221.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 51px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image225.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 48px; width: 353px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image226.png\" \/>\u00a0\u00a0\u00a0\u00a0 \u2026..(i)<\/p>\n<p>Also given Dot product of <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image090.png\" \/>\u00a0is 1.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>.<img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image090.png\" \/>\u00a0= 1<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 336px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image227.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image228.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image229.png\" \/><\/p>\n<p>Squaring both sides,<\/p>\n<p><img decoding=\"async\" style=\"height: 29px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image230.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 151px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image230.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image231.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image232.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image198.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-3\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"14_If_are_mutually_perpendicular_vectors_of_equal_magnitudes_show_that_the_vector_is_equally_inclined_to\"><\/span><strong>14. If <\/strong><img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image048.png\" \/><strong>\u00a0are mutually perpendicular vectors of equal magnitudes, show that the vector <\/strong><img decoding=\"async\" style=\"height: 23px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image233.png\" \/><strong>\u00a0is equally inclined to <\/strong><img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image048.png\" \/><strong>.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image048.png\" \/>\u00a0are mutually perpendicular vectors of equal magnitude.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 25px; width: 247px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image234.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>And <img decoding=\"async\" style=\"height: 32px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image235.png\" \/>\u00a0(say)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p>Let vector <img decoding=\"async\" style=\"height: 23px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image236.png\" \/>\u00a0make angles <img decoding=\"async\" style=\"height: 24px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image237.png\" \/>\u00a0with vectors <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image048.png\" \/>\u00a0respectively.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 61px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image238.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 56px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image239.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 56px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image240.png\" \/>\u00a0\u00a0[From eq. (i)]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image241.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 64px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image242.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 61px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image243.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p>We know that <img decoding=\"async\" style=\"height: 35px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image244.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image245.png\" \/><\/p>\n<p>Putting the values from eq. (i) and (ii),<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 68px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image246.png\" \/>= <img decoding=\"async\" style=\"height: 29px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image247.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image248.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image249.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 24px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image241.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 61px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image243.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 44px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image250.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image251.png\" \/><\/p>\n<p>Similarly, <img decoding=\"async\" style=\"height: 44px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image252.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 44px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image253.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image026.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 24px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image254.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 44px; width: 73px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image255.png\" \/><\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 23px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image233.png\" \/>\u00a0is equally inclined to the vectors <img decoding=\"async\" style=\"height: 25px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image256.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image257.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-4\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"15_Prove_that_if_and_only_if_are_perpendicular_given\"><\/span><strong>15. Prove that <\/strong><img decoding=\"async\" style=\"height: 35px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image258.png\" \/><strong>\u00a0if and only if <\/strong><img decoding=\"async\" style=\"height: 25px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image256.png\" \/><strong>\u00a0are perpendicular given <\/strong><img decoding=\"async\" style=\"height: 25px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image259.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>We know that <img decoding=\"async\" style=\"height: 32px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image260.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image261.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image262.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image263.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Now if <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/>\u00a0are perpendicular\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0\u00a0\u00a0<img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image264.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image264.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 30px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image265.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image263.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image262.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image266.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image267.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image266.png\" \/>\u00a0[Putting value of <img decoding=\"async\" style=\"height: 32px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image262.png\" \/>\u00a0in eq. (i)]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image268.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image264.png\" \/><\/p>\n<p>But <img decoding=\"async\" style=\"height: 25px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image269.png\" \/>\u00a0(given)<\/p>\n<p>Therefore, vectors <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/>\u00a0are perpendicular to each other.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-5\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"16_Choose_the_correct_answer\"><\/span><strong>16. Choose the correct answer:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>If <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0is the angle between two vectors <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 25px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image270.png\" \/>\u00a0then <img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image271.png\" \/>\u00a0only when:<\/strong><\/p>\n<p><strong>(A) <img decoding=\"async\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image272.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 41px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image273.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 19px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image274.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 19px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image275.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image271.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image276.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image277.png\" \/><\/p>\n[<img decoding=\"async\" style=\"height: 32px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image278.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image279.png\" \/>\u00a0being lengths of vectors are always <img decoding=\"async\" style=\"height: 16px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image280.png\" \/>\u00a00]\n<p>Therefore, option (B) is correct.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-6\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"17_Choose_the_correct_answer\"><\/span><strong>17. Choose the correct answer:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Let <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/>\u00a0be two unit vectors and <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0is the angle between them. Then <img decoding=\"async\" style=\"height: 23px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image144.png\" \/>\u00a0is a unit vector if:<\/strong><\/p>\n<p><strong>(A) <img decoding=\"async\" style=\"height: 41px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image281.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 41px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image282.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 41px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image283.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image284.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 25px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image256.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image144.png\" \/>\u00a0are unit vectors.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image285.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image286.png\" \/><\/p>\n<p>Now squaring both sides of <img decoding=\"async\" style=\"height: 32px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image286.png\" \/>, we have,<\/p>\n<p><img decoding=\"async\" style=\"height: 31px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image287.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 31px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image288.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image289.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 33px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image290.png\" \/>\u00a0, where <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/>\u00a0is the given angle between vectors <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/>.<\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 32px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image285.png\" \/>, we have,\u00a0 <img decoding=\"async\" style=\"height: 19px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image291.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image292.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 39px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image293.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image294.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image295.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 38px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image296.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 22px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image297.png\" \/><\/p>\n<p>Therefore, option (D) is correct.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-7\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"18_Choose_the_correct_answer\"><\/span><strong>18. Choose the correct answer:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>The value of <img decoding=\"async\" style=\"height: 35px; width: 185px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image298.png\" \/>\u00a0is:<\/strong><\/p>\n<p><strong>(A) 0\u00a0\u00a0 <\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 17px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image299.png\" \/>\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(C) 1\u00a0\u00a0 <\/strong><\/p>\n<p><strong>(D) 3<\/strong><\/p>\n<p><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 35px; width: 304px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image300.png\" \/><\/p>\n<p>Also <img decoding=\"async\" style=\"height: 27px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image301.png\" \/>\u00a0= 1 \u2013 1 + 1<\/p>\n<p>Therefore, option (C) is correct.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-8\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"19_If_be_the_angle_between_any_two_vectors_and_then_when_is_equal_to\"><\/span><strong>19. If <\/strong><img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/><strong>\u00a0be the angle between any two vectors <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image070.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image056.png\" \/><strong>, then <\/strong><img decoding=\"async\" style=\"height: 32px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image302.png\" \/><strong>\u00a0when <\/strong><img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image176.png\" \/><strong>\u00a0is equal to:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(A) 0\u00a0\u00a0 <\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 41px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image303.png\" \/>\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 41px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image304.png\" \/>\u00a0\u00a0\u00a0\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 15px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image305.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 32px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image306.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image008.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image307.png\" \/><\/p>\n<p>And this equation is true only for option (B) namely <img decoding=\"async\" style=\"height: 36px; width: 38px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image308.png\" \/>, since <img decoding=\"async\" style=\"height: 38px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/10%20Misc\/image309.png\" \/><\/p>\n<p>Therefore, option (B) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Miscellaneous-9\"><\/span>NCERT Solutions class 12 Maths Miscellaneous<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide<\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_App_for_Students\"><\/span>CBSE App for Students<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To download\u00a0NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths &#8230; <a title=\"NCERT Solutions class 12 Maths Miscellaneous\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-miscellaneous-7\/\" aria-label=\"More on NCERT Solutions class 12 Maths Miscellaneous\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":-1,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1432,281],"tags":[283,1342,321,216],"class_list":["post-4528","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-cbse-class-12","category-ncert-solutions","tag-cbse-study-material","tag-class-12","tag-mathematics","tag-ncert-solutions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>NCERT Solutions class 12 Maths Miscellaneous | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Miscellaneous in PDF format for free download. 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