{"id":4525,"date":"2016-05-16T11:49:00","date_gmt":"2016-05-16T06:19:00","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-10-3\/"},"modified":"2018-05-08T17:15:00","modified_gmt":"2018-05-08T11:45:00","slug":"ncert-solutions-class-12-maths-exercise-10-3","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-10-3\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 10.3"},"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-exercise-10-3\/#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-exercise-10-3\/#7_Evaluate_the_product\" >7. Evaluate the product \u00a0<\/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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#8_Find_the_magnitude_of_two_vectors_and_having_the_same_magnitude_such_that_the_angle_between_them_is_and_their_scalar_product_is\" >8. Find the magnitude of two vectors \u00a0and \u00a0having the same magnitude such that the angle between them is \u00a0and their scalar product is \u00a0<\/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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-2\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#9_Find_if_for_a_unit_vector\" >9. Find \u00a0if for a unit vector \u00a0\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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-3\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#10_If_and_are_such_that_is_perpendicular_to_then_find_the_value_of\" >10. If \u00a0and \u00a0are such that \u00a0is perpendicular to \u00a0then 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-9\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-4\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#11_Show_that_is_perpendicular_to_for_any_two_non-zero_vectors_and\" >11. Show that \u00a0is perpendicular to \u00a0for any two non-zero vectors \u00a0and \u00a0<\/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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-5\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#12_If_and_then_what_can_be_concluded_about_the_vector\" >12. If \u00a0and , then what can be concluded about the vector \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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-6\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#13_If_are_unit_vectors_such_that_find_the_value_of\" >13. If \u00a0are unit vectors such that \u00a0find the value of<\/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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-7\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#14_If_either_vector_or_then_But_the_converse_need_not_be_true_Justify_your_answer_with_an_example\" >14. If either vector \u00a0or \u00a0then . But the converse need not be true. Justify your answer with an example.<\/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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-8\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#15_If_the_vertices_A_B_C_of_a_triangle_ABC_are_and_0_1_2_respectively_then_find\" >15. If the vertices A, B, C of a triangle ABC are \u00a0and (0, 1, 2) respectively, then find \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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-9\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#16_Show_that_the_points_A_1_2_7_B_2_6_3_and_C_are_collinear\" >16. Show that the points A (1, 2, 7), B (2, 6, 3) and C\u00a0are collinear.<\/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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-10\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#17_Show_that_the_vectors_and_form_the_vertices_of_a_right_angled_triangle\" >17. Show that the vectors \u00a0and \u00a0form the vertices of a right angled triangle.<\/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-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-11\" >NCERT Solutions class 12 Maths Exercise 10.3<\/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-exercise-10-3\/#18_If_is_a_non-zero_vector_of_magnitude_and_is_a_non-zero_scalar_then_is_a_unit_vector_if\" >18. If \u00a0is a non-zero vector of magnitude \u00a0and \u00a0is a non-zero scalar, then \u00a0is a unit vector if:<\/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-25\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-10-3\/#NCERT_Solutions_class_12_Maths_Exercise_103-12\" >NCERT Solutions class 12 Maths Exercise 10.3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-10-3\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 10.3 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 Exercise 10.3\" width=\"201\" height=\"258\" \/><\/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. Find the angle between two vectors <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/><strong>\u00a0with magnitude <\/strong><img decoding=\"async\" style=\"height: 24px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image003.png\" \/><strong>\u00a0and 2 respectively having <\/strong><img decoding=\"async\" style=\"height: 23px; width: 58px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image004.png\" \/><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 32px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image005.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 24px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image006.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image007.png\" \/>\u00a0be the angle between the vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image008.png\" \/><\/p>\n<p>We know that <img decoding=\"async\" style=\"height: 56px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image009.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image011.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 48px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image012.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image013.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image014.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image015.png\" \/><\/p>\n<hr \/>\n<p><strong>2. Find the angle between the vectors <\/strong><img decoding=\"async\" style=\"height: 25px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image016.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image017.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Let <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image018.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image020.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image021.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 37px; width: 211px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image022.png\" \/><\/p>\n<p>Also <img decoding=\"async\" style=\"height: 23px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image023.png\" \/><\/p>\n<p>= Product of coefficients of <img decoding=\"async\" style=\"height: 24px; width: 11px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image024.png\" \/>\u00a0+ Product of coefficients of <img decoding=\"async\" style=\"height: 27px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image025.png\" \/>\u00a0+ Product of coefficients <img decoding=\"async\" style=\"height: 24px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image026.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 243px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image027.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image007.png\" \/>\u00a0be the angle between the vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image008.png\" \/><\/p>\n<p>We know that <img decoding=\"async\" style=\"height: 56px; width: 88px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image009.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image028.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image029.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image030.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image031.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image032.png\" \/><\/p>\n<hr \/>\n<p><strong>3. Find the projection of the vector <\/strong><img decoding=\"async\" style=\"height: 27px; width: 32px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image033.png\" \/><strong>\u00a0on the vector <\/strong><img decoding=\"async\" style=\"height: 27px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image034.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 27px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image035.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 129px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image036.png\" \/><\/p>\n<p>Projection of vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 56px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image037.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 56px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image038.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 44px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image039.png\" \/><\/p>\n<p>If projection of vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0is zero, then vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0is perpendicular to <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image008.png\" \/><\/p>\n<hr \/>\n<p><strong>4. Find the projection of the vector <\/strong><img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image040.png\" \/><strong>\u00a0on the vector <\/strong><img decoding=\"async\" style=\"height: 27px; width: 72px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image041.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 27px; width: 95px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image042.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image043.png\" \/><\/p>\n<p>Projection of vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 56px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image037.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 56px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image044.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 44px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image045.png\" \/><\/p>\n<hr \/>\n<p><strong>5. Show that each of the given three vectors is a unit vector:<\/strong><\/p>\n<p><strong><img decoding=\"async\" style=\"height: 37px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image046.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 36px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image047.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 36px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image048.png\" \/><\/strong><\/p>\n<p><strong>Also show that they are mutually perpendicular to each other.<\/strong><\/p>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 41px; width: 231px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image049.png\" \/>\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image050.png\" \/>\u00a0\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image051.png\" \/>\u00a0 \u2026\u2026\u2026.(iii)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 53px; width: 387px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image052.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 53px; width: 395px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image053.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 53px; width: 393px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image054.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0Each of the three given vectors <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image056.png\" \/>\u00a0is a unit vector.<\/p>\n<p>From eq. (i) and (ii),<\/p>\n<p><img decoding=\"async\" style=\"height: 45px; width: 245px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image057.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image058.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image059.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0are perpendicular to each other.<\/p>\n<p>From eq. (ii) and eq. (iii),<\/p>\n<p><img decoding=\"async\" style=\"height: 45px; width: 252px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image060.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image058.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0are perpendicular to each other.<\/p>\n<p>From eq. (i) and (iii),<\/p>\n<p><img decoding=\"async\" style=\"height: 45px; width: 244px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image062.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 32px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image058.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 229px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image063.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0are perpendicular to each other.<\/p>\n<p>Hence, <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image056.png\" \/>\u00a0are mutually perpendicular vectors.<\/p>\n<hr \/>\n<p><strong>6. Find <\/strong><img decoding=\"async\" style=\"height: 32px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image064.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 32px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image065.png\" \/><strong>\u00a0if <\/strong><img decoding=\"async\" style=\"height: 30px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image066.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 30px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image067.png\" \/><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 32px; width: 119px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image068.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image069.png\" \/>\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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image070.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image071.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image072.png\" \/>\u00a0\u00a0\u00a0 \u2026\u2026\u2026.(ii)<\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 32px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image069.png\" \/>\u00a0in eq. (ii),<\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image073.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image074.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image075.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image076.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 48px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image077.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 48px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image078.png\" \/>\u00a0in eq (i),<\/p>\n<p><img decoding=\"async\" style=\"height: 53px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image079.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"7_Evaluate_the_product\"><\/span><strong>7. Evaluate the product <\/strong><img decoding=\"async\" style=\"height: 29px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image080.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 32px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image081.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 32px; width: 297px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image082.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 23px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image083.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 180px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image084.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image085.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"8_Find_the_magnitude_of_two_vectors_and_having_the_same_magnitude_such_that_the_angle_between_them_is_and_their_scalar_product_is\"><\/span><strong>8. Find the magnitude of two vectors <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 25px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image086.png\" \/><strong>\u00a0having the same magnitude such that the angle between them is <\/strong><img decoding=\"async\" style=\"height: 21px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image087.png\" \/><strong>\u00a0and their scalar product is <\/strong><img decoding=\"async\" style=\"height: 41px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image088.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 32px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image089.png\" \/>, angle <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image007.png\" \/>\u00a0(say) between <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0is <img decoding=\"async\" style=\"height: 21px; width: 25px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image087.png\" \/>\u00a0and their scalar (i.e., dot) product = <img decoding=\"async\" style=\"height: 41px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image090.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image091.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image092.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 32px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image089.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 24px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image093.png\" \/>\u00a0we have <img decoding=\"async\" style=\"height: 41px; width: 112px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image094.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 84px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image095.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image096.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image097.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image098.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image097.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image097.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image099.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-2\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"9_Find_if_for_a_unit_vector\"><\/span><strong>9. Find <\/strong><img decoding=\"async\" style=\"height: 32px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image100.png\" \/><strong>\u00a0if for a unit vector <\/strong><img decoding=\"async\" style=\"height: 25px; width: 17px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image101.png\" \/><strong>\u00a0<\/strong><img decoding=\"async\" style=\"height: 29px; width: 122px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image102.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0is a unit vector <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image097.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 32px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image103.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image104.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image105.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 157px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image106.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image107.png\" \/><\/p>\n<p>Putting <img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image097.png\" \/>\u00a0from eq. (i), <img decoding=\"async\" style=\"height: 35px; width: 75px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image108.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 55px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image109.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image110.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-3\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"10_If_and_are_such_that_is_perpendicular_to_then_find_the_value_of\"><\/span><strong>10. If <\/strong><img decoding=\"async\" style=\"height: 27px; width: 200px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image111.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image112.png\" \/><strong>\u00a0are such that <\/strong><img decoding=\"async\" style=\"height: 23px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image113.png\" \/><strong>\u00a0is perpendicular to <\/strong><img decoding=\"async\" style=\"height: 25px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image114.png\" \/><strong>\u00a0then 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\/Ex10.3\/image115.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image116.png\" \/>, <img decoding=\"async\" style=\"height: 27px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image117.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image112.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 35px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image118.png\" \/>\u00a0=<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 29px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image120.png\" \/><\/p>\n<p>Again, <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image112.png\" \/>= <img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image121.png\" \/><\/p>\n<p>Since, <img decoding=\"async\" style=\"height: 32px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image122.png\" \/>\u00a0is perpendicular to <img decoding=\"async\" style=\"height: 25px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image114.png\" \/>\u00a0therefore, <img decoding=\"async\" style=\"height: 32px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image122.png\" \/>.<img decoding=\"async\" style=\"height: 23px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image125.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image126.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image127.png\" \/>\u00a0\u00a0 <img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 19px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image128.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-4\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"11_Show_that_is_perpendicular_to_for_any_two_non-zero_vectors_and\"><\/span><strong>11. Show that <\/strong><img decoding=\"async\" style=\"height: 27px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image129.png\" \/><strong>\u00a0is perpendicular to <\/strong><img decoding=\"async\" style=\"height: 28px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image130.png\" \/><strong>\u00a0for any two non-zero vectors <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image008.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 32px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image131.png\" \/>, where <img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image132.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image133.png\" \/><\/p>\n<p>Let <img decoding=\"async\" style=\"height: 32px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image134.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 32px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image135.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 23px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image136.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image137.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 92px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image138.png\" \/><\/p>\n<p>Putting, <img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image132.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image133.png\" \/>,<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image139.png\" \/>\u00a0= 0<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image140.png\" \/><\/p>\n<p>Therefore, vectors <img decoding=\"async\" style=\"height: 23px; width: 12px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image141.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image142.png\" \/>\u00a0are perpendicular ot each other.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-5\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"12_If_and_then_what_can_be_concluded_about_the_vector\"><\/span><strong>12. If <\/strong><img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image143.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image144.png\" \/><strong>, then what can be concluded about the vector <\/strong><img decoding=\"async\" style=\"height: 23px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image145.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\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\/Ex10.3\/image143.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image146.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image147.png\" \/><\/p>\n<p>Again <img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image144.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image148.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image149.png\" \/>\u00a0\u00a0<img decoding=\"async\" style=\"height: 35px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image150.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a00 = 0 for all (any vector <img decoding=\"async\" style=\"height: 23px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image008.png\" \/>)<\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image002.png\" \/>\u00a0can be any vector.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-6\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"13_If_are_unit_vectors_such_that_find_the_value_of\"><\/span><strong>13. If <\/strong><img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image056.png\" \/><strong>\u00a0are unit vectors such that <\/strong><img decoding=\"async\" style=\"height: 21px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image151.png\" \/><strong>\u00a0find the value of <\/strong><img decoding=\"async\" style=\"height: 21px; width: 82px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image152.png\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Since, <img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image056.png\" \/>\u00a0are unit vectors.<\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 32px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image153.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image154.png\" \/>\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Also given <img decoding=\"async\" style=\"height: 23px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image156.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 257px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image157.png\" \/>\u00a0 <img decoding=\"async\" style=\"height: 35px; width: 305px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image158.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 237px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image159.png\" \/><\/p>\n<p>Putting the values from eq. (i), we get<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 192px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image161.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image162.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-7\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"14_If_either_vector_or_then_But_the_converse_need_not_be_true_Justify_your_answer_with_an_example\"><\/span><strong>14. If either vector <\/strong><img decoding=\"async\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image163.png\" \/><strong>\u00a0or <\/strong><img decoding=\"async\" style=\"height: 25px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image164.png\" \/><strong>\u00a0then <\/strong><img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image144.png\" \/><strong>. But the converse need not be true. Justify your answer with an example.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong><strong>Case <\/strong>I: Vector <img decoding=\"async\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image163.png\" \/>. Therefore by definition of zero vector, <img decoding=\"async\" style=\"height: 32px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image147.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image165.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 32px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image166.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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image144.png\" \/><\/p>\n<p><strong>Case II<\/strong>: Vector <img decoding=\"async\" style=\"height: 23px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image167.png\" \/>. Therefore by definition of zero vector, <img decoding=\"async\" style=\"height: 32px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image168.png\" \/>\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\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image165.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 23px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image169.png\" \/>\u00a0\u00a0[From eq. (ii)]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image144.png\" \/><\/p>\n<p>But the converse is not true.<\/p>\n<p><strong>Justification<\/strong>: Let <img decoding=\"async\" style=\"height: 27px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image170.png\" \/><\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 36px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image171.png\" \/><\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image172.png\" \/><\/p>\n<p>Again let <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image173.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 36px; width: 219px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image174.png\" \/><\/p>\n<p>Therefore, <img decoding=\"async\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image175.png\" \/><\/p>\n<p>But <img decoding=\"async\" style=\"height: 28px; width: 248px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image176.png\" \/><\/p>\n<p>Hence, here <img decoding=\"async\" style=\"height: 23px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image144.png\" \/>, but <img decoding=\"async\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image172.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 37px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image175.png\" \/>.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-8\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"15_If_the_vertices_A_B_C_of_a_triangle_ABC_are_and_0_1_2_respectively_then_find\"><\/span><strong>15. If the vertices A, B, C of a triangle ABC are <\/strong><img decoding=\"async\" style=\"height: 27px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image177.png\" \/><strong>\u00a0and (0, 1, 2) respectively, then find <\/strong><img decoding=\"async\" style=\"height: 19px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image178.png\" \/><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Vertices A, B, C of a triangle are A (1, 2,3 ), B<img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image179.png\" \/>\u00a0and C (0, 1, 2) respectively.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0Position vector of point A = <img decoding=\"async\" style=\"height: 29px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image180.png\" \/><\/p>\n<p>Position vector of point B = <img decoding=\"async\" style=\"height: 29px; width: 188px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image181.png\" \/><\/p>\n<p>Position vector of point C = <img decoding=\"async\" style=\"height: 29px; width: 175px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image182.png\" \/><\/p>\n<p>Now <img decoding=\"async\" style=\"height: 21px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image183.png\" \/>\u00a0= Position vector of point A \u2013 Position vector of point B<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image184.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image185.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>And <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image186.png\" \/>\u00a0= Position vector of point C \u2013 Position vector of point B<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image187.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 224px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image188.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image007.png\" \/>\u00a0be the angle between the vectors <img decoding=\"async\" style=\"height: 21px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image183.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image186.png\" \/>.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 56px; width: 116px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image189.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 47px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image190.png\" \/>\u00a0\u00a0[Using eq. (i) and (ii)]\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image191.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image192.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-9\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"16_Show_that_the_points_A_1_2_7_B_2_6_3_and_C_are_collinear\"><\/span><strong>16. Show that the points A (1, 2, 7), B (2, 6, 3) and C<\/strong><img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image193.png\" \/><strong>\u00a0are collinear.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Vertices A, B, C of a triangle are A (1, 2, 7), B (2, 6, 3) and C<img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image193.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\/Ex10.3\/image055.png\" \/>\u00a0Position vector of point A = <img decoding=\"async\" style=\"height: 29px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image194.png\" \/><\/p>\n<p>Position vector of point B = <img decoding=\"async\" style=\"height: 29px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image195.png\" \/><\/p>\n<p>Position vector of point C = <img decoding=\"async\" style=\"height: 29px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image196.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\/Ex10.3\/image197.png\" \/>\u00a0= Position vector of point B \u2013 Position vector of point A<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image198.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image199.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>And <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image200.png\" \/>\u00a0= Position vector of point C \u2013 Position vector of point A<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image201.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 348px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image202.png\" \/>\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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image200.png\" \/>\u00a0= 2.<img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image197.png\" \/>\u00a0\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\/Ex10.3\/image010.png\" \/>\u00a0Vectors <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image197.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image200.png\" \/>\u00a0are collinear and parallel. <img decoding=\"async\" style=\"height: 32px; width: 76px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image203.png\" \/><\/p>\n<p>Thus, points A, B and C are collinear.<\/p>\n<p>And also vectors <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image197.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image200.png\" \/>\u00a0have a common point A and hence can\u2019t be parallel.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-10\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"17_Show_that_the_vectors_and_form_the_vertices_of_a_right_angled_triangle\"><\/span><strong>17. Show that the vectors <\/strong><img decoding=\"async\" style=\"height: 24px; width: 127px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image204.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 23px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image205.png\" \/><strong>\u00a0form the vertices of a right angled triangle.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let the given position vectors be A, B, C.<\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0Position vector of point A is <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image206.png\" \/>\u00a0Position vector of point B is <img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image207.png\" \/>\u00a0and Position vector of point C is <img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image208.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image055.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image197.png\" \/>\u00a0= Position vector of B \u2013 Position vector of A<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 153px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image209.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 141px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image210.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image211.png\" \/>\u00a0\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image186.png\" \/>\u00a0= Position vector of C \u2013 Position vector of B<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image212.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image213.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image214.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(ii)<\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image200.png\" \/>\u00a0= Position vector of C \u2013 Position vector of A<\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image215.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image216.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image207.png\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u2026\u2026\u2026.(iii)<\/p>\n<p>Adding eq. (i) and (ii),<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image197.png\" \/>\u00a0+ <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image186.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image211.png\" \/>\u00a0+ <img decoding=\"async\" style=\"height: 27px; width: 63px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image214.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image207.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image200.png\" \/>\u00a0[Using eq. (iii)]\n<p>Therefore, by Triangle law of addition of vectors, points A, B, C are the vertices of a triangle ABC.<\/p>\n<p>Now from eq. (i) and (ii),<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image197.png\" \/>.<img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image186.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 191px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image217.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 123px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image218.png\" \/><\/p>\n<p>Again from eq. (ii) and (iii),<\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image186.png\" \/>.<img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image200.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 179px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image219.png\" \/>\u00a0= 2 + 3 \u2013 5 = 0<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image186.png\" \/>\u00a0is perpendicular to <img decoding=\"async\" style=\"height: 23px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0Angle C is <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image220.png\" \/>\u00a0Therefore <img decoding=\"async\" style=\"height: 19px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image221.png\" \/>\u00a0is a right angled at C.<\/p>\n<p>Thus, A, B, C are the vertices of a right angled triangle.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-11\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"18_If_is_a_non-zero_vector_of_magnitude_and_is_a_non-zero_scalar_then_is_a_unit_vector_if\"><\/span><strong>18. If <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/><strong>\u00a0is a non-zero vector of magnitude <\/strong><img decoding=\"async\" style=\"height: 19px; width: 21px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image222.png\" \/><strong>\u00a0and <\/strong><img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image223.png\" \/><strong>\u00a0is a non-zero scalar, then <\/strong><img decoding=\"async\" style=\"height: 23px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image224.png\" \/><strong>\u00a0is a unit vector if:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(A) <img decoding=\"async\" style=\"height: 19px; width: 36px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image225.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p><strong>(B) <img decoding=\"async\" style=\"height: 19px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image226.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p><strong>(C) <img decoding=\"async\" style=\"height: 27px; width: 44px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image227.png\" \/>\u00a0\u00a0<\/strong><\/p>\n<p><strong>(D) <img decoding=\"async\" style=\"height: 47px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image228.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image001.png\" \/>\u00a0is a non-zero vector of magnitude <img decoding=\"async\" style=\"height: 15px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image229.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image097.png\" \/><\/p>\n<p>Also given <img decoding=\"async\" style=\"height: 19px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image230.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 24px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image224.png\" \/>\u00a0is a unit vector.<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/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\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image232.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image233.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image010.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 47px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex10.3\/image228.png\" \/><\/p>\n<p>Therefore, option (D) is correct.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_103-12\"><\/span>NCERT Solutions class 12 Maths Exercise 10.3<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 Exercise 10.3 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 &#8230; <a title=\"NCERT Solutions class 12 Maths Exercise 10.3\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-10-3\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 10.3\">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-4525","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 Exercise 10.3 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 10.3 in PDF format for free download. 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