{"id":4536,"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-11-2\/"},"modified":"2018-05-08T15:40:38","modified_gmt":"2018-05-08T10:10:38","slug":"ncert-solutions-class-12-maths-exercise-11-2","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-2\/","title":{"rendered":"NCERT Solutions class 12 Maths Exercise 11.2"},"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-11-2\/#NCERT_Solutions_class_12_Maths_Three_Dimensional_Geometry\" >NCERT Solutions class 12 Maths Three Dimensional Geometry\u00a0<\/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-11-2\/#2Show_that_the_line_through_the_points_is_perpendicular_to_the_line_through_the_points_0_3_2_and_3_5_6\" >2.Show that the line through the points \u00a0is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).<\/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-11-2\/#3Show_that_the_line_through_points_4_7_8_2_3_4_is_parallel_to_the_line_through_the_points\" >3.Show that the line through points (4, 7, 8), (2, 3, 4) is parallel to the line through the points \u00a0<\/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-11-2\/#4Find_the_equation_of_the_line_which_passes_through_the_point_1_2_3_and_is_parallel_to_the_vector\" >4.Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-2\/#5Find_the_equation_of_the_line_in_vector_and_in_Cartesian_form_that_passes_through_the_point_with_position_vector_and_is_in_the_direction\" >5.Find the equation of the line in vector and in Cartesian form that passes through the point with position vector \u00a0and is in the direction \u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-2\/#6Find_the_Cartesian_equation_of_the_line_which_passes_through_the_point_and_parallel_to_the_line_given_by\" >6.Find the Cartesian equation of the line which passes through the point \u00a0and parallel to the line given by \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-11-2\/#7The_Cartesian_equation_of_a_line_is_Write_its_vector_form\" >7.The Cartesian equation of a line is \u00a0Write its vector form.<\/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-11-2\/#8Find_the_vector_and_Cartesian_equations_of_the_line_that_passes_through_the_origin_and\" >8.Find the vector and Cartesian equations of the line that passes through the origin and \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-11-2\/#9Find_vector_and_Cartesian_equations_of_the_line_that_passes_through_the_points_and\" >9.Find vector and Cartesian equations of the line that passes through the points \u00a0and \u00a0<\/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-11-2\/#10Find_the_angle_between_the_following_pairs_of_lines\" >10.Find the angle between the following pairs of lines:<\/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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#11Find_the_angle_between_the_following_pair_of_lines\" >11.Find the angle between the following pair of lines:<\/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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112-2\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#12Find_the_values_of_so_that_the_lines_and_are_at_right_angles\" >12.Find the values of \u00a0so that the lines \u00a0and \u00a0are at right angles.<\/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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112-3\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#13Show_that_the_lines_and_are_perpendicular_to_each_other\" >13.Show that the lines \u00a0and \u00a0are perpendicular to each other.<\/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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112-4\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#14Find_the_shortest_distance_between_the_lines_and\" >14.Find the shortest distance between the lines \u00a0and \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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112-5\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#15Find_the_shortest_distance_between_the_lines_and\" >15.Find the shortest distance between the lines \u00a0and .<\/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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112-6\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#16Find_the_shortest_distance_between_the_lines_whose_vector_equations_are\" >16.Find the shortest distance between the lines whose vector equations are<\/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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112-7\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#17Find_the_shortest_distance_between_the_lines_whose_vector_equations_are\" >17.Find the shortest distance between the lines whose vector equations are<\/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-11-2\/#NCERT_Solutions_class_12_Maths_Exercise_112-8\" >NCERT Solutions class 12 Maths Exercise 11.2<\/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-11-2\/#CBSE_App_for_Students\" >CBSE App for Students<\/a><\/li><\/ul><\/nav><\/div>\n<p>NCERT Solutions class 12 Maths Exercise 11.2 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-three-dimensional-geometry\/1295\/ncert-solutions\/5\/\">NCERT solutions for\u00a0Three Dimensional Geometry\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 11.2\" width=\"182\" height=\"233\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Three_Dimensional_Geometry\"><\/span>NCERT Solutions class 12 Maths Three Dimensional Geometry<strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>.Show that the three lines with direction cosines <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 235px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image001.png\" \/><\/strong><strong>\u00a0are mutually perpendicular.<\/strong><\/p>\n<p><strong>Ans. <\/strong>Given: Direction cosines of three lines are<\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image002.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image003.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image004.png\" \/><\/p>\n<p>For first two lines,<img decoding=\"async\" style=\"height: 45px; width: 359px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image005.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image006.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image007.png\" \/><\/p>\n<p>Therefore, the first two lines are perpendicular to each other.<\/p>\n<p>For second and third lines,<img decoding=\"async\" style=\"height: 45px; width: 359px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image008.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image009.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image010.png\" \/><\/p>\n<p>Therefore, second and third lines are also perpendicular to each other.<\/p>\n<p>For First and third lines,<img decoding=\"async\" style=\"height: 45px; width: 360px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image011.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 108px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image012.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 145px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image013.png\" \/><\/p>\n<p>Therefore, first and third lines are also perpendicular to each other.<\/p>\n<p>Hence, given three lines are mutually perpendicular to each other.<\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"2Show_that_the_line_through_the_points_is_perpendicular_to_the_line_through_the_points_0_3_2_and_3_5_6\"><\/span><strong>2.Show that the line through the points <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 120px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image014.png\" \/><\/strong><strong>\u00a0is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>We know that direction ratios of the line joining the points A<img decoding=\"async\" style=\"height: 27px; width: 57px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image015.png\" \/>\u00a0and B<img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image016.png\" \/>\u00a0are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image017.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 24px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image020.png\" \/><\/p>\n<p>Again, direction ratios of the line joining the points C (0, 3, 2) and D (3, 5, 6) are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image017.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image021.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 24px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image022.png\" \/>\u00a0(say)<\/p>\n<p>For lines AB and CD,<img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image023.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image024.png\" \/>\u00a0= 6 + 10 \u2013 16 = 0<\/p>\n<p>Therefore, line AB is perpendicular to line CD.<\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"3Show_that_the_line_through_points_4_7_8_2_3_4_is_parallel_to_the_line_through_the_points\"><\/span><strong>3.Show that the line through points (4, 7, 8), (2, 3, 4) is parallel to the line through the points <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image025.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans.\u00a0 <\/strong>We know that direction ratios of the line joining the points A (4, 7, 8) and B (2, 3, 4) are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image017.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image026.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image027.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image028.png\" \/>\u00a0(say)<\/p>\n<p>Again direction ratios of the line joining the points C<img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image029.png\" \/>\u00a0and D (1, 2, 5) are <img decoding=\"async\" style=\"height: 24px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image017.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 27px; width: 144px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 40px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image031.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image032.png\" \/>\u00a0(say)<\/p>\n<p>For the lines AB and CD,<img decoding=\"async\" style=\"height: 45px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image033.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 45px; width: 85px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image035.png\" \/><\/p>\n<p>Therefore, line AB is parallel to line CD.<\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"4Find_the_equation_of_the_line_which_passes_through_the_point_1_2_3_and_is_parallel_to_the_vector\"><\/span><strong>4.Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image036.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans.\u00a0 <\/strong>A point on the required line is A (1, 2, 3) = <img decoding=\"async\" style=\"height: 24px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image037.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0Position vector of a point on the required line is <img decoding=\"async\" style=\"height: 29px; width: 193px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image038.png\" \/><\/p>\n<p>The required line is parallel to the vector <img decoding=\"async\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image039.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0direction ratios of the required line are coefficient of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image040.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image041.png\" \/>\u00a0are<\/p>\n<p><img decoding=\"async\" style=\"height: 21px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image042.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>Vector equation of the required line is<\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image043.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image044.png\" \/><\/p>\n<p>Where <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image045.png\" \/>\u00a0is a real number.<\/p>\n<p>Cartesian equation of this equation is <img decoding=\"async\" style=\"height: 41px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image046.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"5Find_the_equation_of_the_line_in_vector_and_in_Cartesian_form_that_passes_through_the_point_with_position_vector_and_is_in_the_direction\"><\/span><strong>5.Find the equation of the line in vector and in Cartesian form that passes through the point with position vector <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image047.png\" \/><\/strong><strong>\u00a0and is in the direction <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 65px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image048.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Position vector of a point on the required line is <img decoding=\"async\" style=\"height: 27px; width: 96px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image049.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image050.png\" \/><\/p>\n<p>The required line is in the direction of the vector is <img decoding=\"async\" style=\"height: 27px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image051.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0Direction ratios of required line are coefficients of <img decoding=\"async\" style=\"height: 27px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image040.png\" \/>\u00a0in <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image041.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image052.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image053.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Equation of the required line in vector form is <img decoding=\"async\" style=\"height: 23px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image043.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image054.png\" \/><\/p>\n<p>Where <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image045.png\" \/>\u00a0is a real number.<\/p>\n<p>Cartesian equation of this equation is <img decoding=\"async\" style=\"height: 41px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image055.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"6Find_the_Cartesian_equation_of_the_line_which_passes_through_the_point_and_parallel_to_the_line_given_by\"><\/span><strong>6.Find the Cartesian equation of the line which passes through the point <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image056.png\" \/><\/strong><strong>\u00a0and parallel to the line given by <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image057.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: A point on the line is <img decoding=\"async\" style=\"height: 27px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image058.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image059.png\" \/><\/p>\n<p>Equation of the given line in Cartesian form is <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image060.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Direction ratios of the given line are its denominators 3, 5, 6 <img decoding=\"async\" style=\"height: 21px; width: 52px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image053.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>Equation of the required line is <img decoding=\"async\" style=\"height: 41px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image061.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 44px; width: 177px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image062.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image063.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"7The_Cartesian_equation_of_a_line_is_Write_its_vector_form\"><\/span><strong>7.The Cartesian equation of a line is <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 139px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image064.png\" \/><\/strong><strong>\u00a0Write its vector form.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: The Cartesian equation of the line is <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image065.png\" \/>= <img decoding=\"async\" style=\"height: 19px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image045.png\" \/>\u00a0(say)<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 223px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image066.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 21px; width: 232px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image067.png\" \/><\/p>\n<p>General equation for the required line is <img decoding=\"async\" style=\"height: 27px; width: 104px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image068.png\" \/><\/p>\n<p>Putting the values of <img decoding=\"async\" style=\"height: 17px; width: 41px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image069.png\" \/>\u00a0in this equation,<\/p>\n<p><img decoding=\"async\" style=\"height: 29px; width: 248px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image070.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 225px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image072.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 32px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image073.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"8Find_the_vector_and_Cartesian_equations_of_the_line_that_passes_through_the_origin_and\"><\/span><strong>8.Find the vector and Cartesian equations of the line that passes through the origin and <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image074.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image075.png\" \/>\u00a0= Position vector of a point here O (say) on the line = (0, 0, 0) = <img decoding=\"async\" style=\"height: 27px; width: 103px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image076.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image041.png\" \/>\u00a0= A vector along the line<\/p>\n<p>= <img decoding=\"async\" style=\"height: 23px; width: 28px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image077.png\" \/>\u00a0= Position vector of point A \u2013 Position vector of point O<\/p>\n<p>= <img decoding=\"async\" style=\"height: 29px; width: 280px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image078.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Vector equation of the line is <img decoding=\"async\" style=\"height: 32px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 35px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image079.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image080.png\" \/><\/p>\n<p>NowCartesian equation of the line<\/p>\n<p>Direction ratios of line OA are <img decoding=\"async\" style=\"height: 21px; width: 171px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image081.png\" \/><\/p>\n<p>And a point on the line is O (0, 0, 0) = <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image034.png\" \/>Cartesian equation of the line = <img decoding=\"async\" style=\"height: 41px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image061.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image082.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image083.png\" \/><\/p>\n<p>Remark:In the solution of the above question we can also take:<\/p>\n<p><img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image075.png\" \/>\u00a0= Position vector of point A = <img decoding=\"async\" style=\"height: 29px; width: 148px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image084.png\" \/>\u00a0for vector form and point A as <img decoding=\"async\" style=\"height: 27px; width: 67px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image059.png\" \/>= <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image085.png\" \/>\u00a0for Cartesian form.<\/p>\n<p>Then the equation of the line in vector form is <img decoding=\"async\" style=\"height: 35px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image086.png\" \/><\/p>\n<p>And equation of line in Cartesian form is <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image087.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"9Find_vector_and_Cartesian_equations_of_the_line_that_passes_through_the_points_and\"><\/span><strong>9.Find vector and Cartesian equations of the line that passes through the points <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image088.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 27px; width: 64px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image089.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Let <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image075.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image041.png\" \/>\u00a0be the position vectors of the points A<img decoding=\"async\" style=\"height: 27px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image088.png\" \/>\u00a0and B<img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image090.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\/Ex11.2\/image034.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image091.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image092.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>A vector along the line = <img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image093.png\" \/>\u00a0= Position vector of point B \u2013 Position vector of point A<\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 21px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image093.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 23px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image094.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image095.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 11px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image096.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image097.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image098.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image099.png\" \/>Vector equation of the line is <img decoding=\"async\" style=\"height: 32px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image073.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/><img decoding=\"async\" style=\"height: 35px; width: 160px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image100.png\" \/><\/p>\n<p>And another vector equation for the same line is <img decoding=\"async\" style=\"height: 23px; width: 83px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image101.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 161px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image102.png\" \/><\/p>\n<p>Cartesian equation<\/p>\n<p>Direction ratios of line AB are <img decoding=\"async\" style=\"height: 21px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image103.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>Equation of the line is <img decoding=\"async\" style=\"height: 41px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image061.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image104.png\" \/><\/p>\n<hr \/>\n<h4><span class=\"ez-toc-section\" id=\"10Find_the_angle_between_the_following_pairs_of_lines\"><\/span><strong>10.Find the angle between the following pairs of lines:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(i)<\/strong><strong><img decoding=\"async\" style=\"height: 35px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image105.png\" \/><\/strong><strong> and <\/strong><strong><img decoding=\"async\" style=\"height: 35px; width: 169px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image106.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>(ii)<\/strong><strong><img decoding=\"async\" style=\"height: 35px; width: 189px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image107.png\" \/><\/strong><strong> and <\/strong><strong><img decoding=\"async\" style=\"height: 35px; width: 213px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image108.png\" \/><\/strong><\/p>\n<p><strong>Ans. <\/strong>(i)Equation of the first line is <img decoding=\"async\" style=\"height: 35px; width: 205px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image105.png\" \/><\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 32px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image073.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image109.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image110.png\" \/><\/p>\n<p>(vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image075.png\" \/>\u00a0is the position vector of a point on line and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image041.png\" \/>\u00a0is a vector along the line)<\/p>\n<p>Again, equation of the second line is <img decoding=\"async\" style=\"height: 35px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image111.png\" \/><\/p>\n<p>Comparing with <img decoding=\"async\" style=\"height: 32px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image112.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 79px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image113.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 100px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image114.png\" \/><\/p>\n<p>(vector <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image075.png\" \/>\u00a0is the position vector of a point on line and <img decoding=\"async\" style=\"height: 23px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image041.png\" \/>\u00a0is a vector along the line)<\/p>\n<p>Let <img decoding=\"async\" style=\"height: 19px; width: 13px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image115.png\" \/>\u00a0be the angle between these two lines, then<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image116.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 47px; width: 217px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image117.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 35px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image118.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 41px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image119.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image120.png\" \/><\/p>\n<p>(ii)Comparing the first and second equations with <img decoding=\"async\" style=\"height: 32px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image073.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 32px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image112.png\" \/>\u00a0resp.<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 89px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image121.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 107px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image122.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\/Ex11.2\/image115.png\" \/>\u00a0be the angle between these two lines, then<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image116.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 47px; width: 249px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image123.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image124.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 44px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image125.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 93px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image126.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"11Find_the_angle_between_the_following_pair_of_lines\"><\/span><strong>11.Find the angle between the following pair of lines:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>(i)<\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image127.png\" \/><\/strong><strong> and <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image128.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>(ii)<\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image129.png\" \/><\/strong><strong> and <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image130.png\" \/><\/strong><strong>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>(i)Given: Equation of first line is <img decoding=\"async\" style=\"height: 41px; width: 132px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image127.png\" \/><\/p>\n<p>The direction ratios of this line i.e., a vector along the line is<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image131.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 60px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image132.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image133.png\" \/><\/p>\n<p>Nowequation of second line is <img decoding=\"async\" style=\"height: 41px; width: 135px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image128.png\" \/><\/p>\n<p>The direction ratios of this line i.e., a vector along the line is<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image134.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 59px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image135.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image136.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\/Ex11.2\/image115.png\" \/>\u00a0be the angle between these two lines, then<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image116.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 47px; width: 258px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image137.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 44px; width: 43px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image138.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 44px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image139.png\" \/><\/p>\n<p>(ii)Given: Equation of first line is <img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image129.png\" \/><\/p>\n<p>The direction ratios of this line i.e., a vector along the line is<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image131.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image140.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image141.png\" \/><\/p>\n<p>Nowequation of second line is <img decoding=\"async\" style=\"height: 41px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image130.png\" \/><\/p>\n<p>The direction ratios of this line i.e., a vector along the line is<\/p>\n<p><img decoding=\"async\" style=\"height: 27px; width: 19px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image134.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image142.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 71px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image143.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\/Ex11.2\/image115.png\" \/>\u00a0be the angle between these two lines, then<\/p>\n<p><img decoding=\"async\" style=\"height: 56px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image116.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 47px; width: 213px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image144.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 41px; width: 62px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image145.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 41px; width: 74px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image146.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112-2\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"12Find_the_values_of_so_that_the_lines_and_are_at_right_angles\"><\/span><strong>12.Find the values of <\/strong><strong><img decoding=\"async\" style=\"height: 17px; width: 16px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image147.png\" \/><\/strong><strong>\u00a0so that the lines <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image148.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 44px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image149.png\" \/><\/strong><strong>\u00a0are at right angles.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Given: Equation of one line <img decoding=\"async\" style=\"height: 44px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image148.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 47px; width: 177px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image018.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 64px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image151.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Direction ratios of this line are <img decoding=\"async\" style=\"height: 41px; width: 125px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image152.png\" \/><\/p>\n<p>Again, equation of another line <img decoding=\"async\" style=\"height: 44px; width: 143px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image149.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 47px; width: 187px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image153.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 61px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image154.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Direction ratios of this line are <img decoding=\"async\" style=\"height: 41px; width: 136px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image155.png\" \/><\/p>\n<p>Since, these two lines are perpendicular.<\/p>\n<p>Therefore,<img decoding=\"async\" style=\"height: 24px; width: 131px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 45px; width: 240px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image157.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 61px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image159.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 41px; width: 49px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image160.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112-3\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"13Show_that_the_lines_and_are_perpendicular_to_each_other\"><\/span><strong>13.Show that the lines <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image161.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image162.png\" \/><\/strong><strong>\u00a0are perpendicular to each other.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Equation of one line <img decoding=\"async\" style=\"height: 41px; width: 113px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image161.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Direction ratios of this line are <img decoding=\"async\" style=\"height: 21px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image163.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 51px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image028.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 28px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image164.png\" \/><\/p>\n<p>Again equation of another line <img decoding=\"async\" style=\"height: 41px; width: 69px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image162.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Direction ratios of this line are 1, 2, 3 = <img decoding=\"async\" style=\"height: 24px; width: 56px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image032.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 16px; width: 20px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image018.png\" \/><img decoding=\"async\" style=\"height: 28px; width: 111px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image165.png\" \/><\/p>\n<p>Now<img decoding=\"async\" style=\"height: 27px; width: 45px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image166.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image023.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 27px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image167.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 87px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image168.png\" \/><\/p>\n<p>Hence, the given two lines are perpendicular to each other.<\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112-4\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"14Find_the_shortest_distance_between_the_lines_and\"><\/span><strong>14.Find the shortest distance between the lines <\/strong><strong><img decoding=\"async\" style=\"height: 35px; width: 175px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image169.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 35px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image170.png\" \/><\/strong><strong>\u00a0<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Comparing the given equations with <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image171.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 27px; width: 80px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image172.png\" \/>, we get<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 196px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image173.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 215px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image174.png\" \/><\/p>\n<p>Since, the shortest distance between the two skew lines is given by<\/p>\n<p><img decoding=\"async\" style=\"height: 64px; width: 147px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image175.png\" \/>\u00a0\u2026\u2026\u2026.(i)<\/p>\n<p>Here,<img decoding=\"async\" style=\"height: 35px; width: 296px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image176.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 83px; width: 396px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image177.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 36px; width: 277px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image178.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 35px; width: 489px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image179.png\" \/><\/p>\n<p>Putting these values in eq. (i),<\/p>\n<p>Shortest distance <img decoding=\"async\" style=\"height: 48px; width: 203px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image180.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112-5\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"15Find_the_shortest_distance_between_the_lines_and\"><\/span><strong>15.Find the shortest distance between the lines <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image181.png\" \/><\/strong><strong>\u00a0and <\/strong><strong><img decoding=\"async\" style=\"height: 41px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image182.png\" \/><\/strong><strong>.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong>Ans. <\/strong>Equation of one line is <img decoding=\"async\" style=\"height: 41px; width: 128px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image181.png\" \/><\/p>\n<p>Comparing this equation with <img decoding=\"async\" style=\"height: 45px; width: 149px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image183.png\" \/>, we have<\/p>\n<p><img decoding=\"async\" style=\"height: 24px; width: 319px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image184.png\" \/><\/p>\n<p>Again equation of another line is <img decoding=\"async\" style=\"height: 41px; width: 133px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image182.png\" \/><\/p>\n<p>Comparing this equation with <img decoding=\"async\" style=\"height: 45px; width: 155px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image185.png\" \/>, we have<\/p>\n<p><img decoding=\"async\" style=\"height: 24px; width: 303px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/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\/Ex11.2\/image034.png\" \/>\u00a0<img decoding=\"async\" style=\"height: 75px; width: 172px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image187.png\" \/>\u00a0 = <img decoding=\"async\" style=\"height: 75px; width: 124px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image188.png\" \/><\/p>\n<p>Expanding by first row = <img decoding=\"async\" style=\"height: 27px; width: 212px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image189.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 19px; width: 137px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image190.png\" \/><\/p>\n<p>And<img decoding=\"async\" style=\"height: 31px; width: 271px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image191.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 35px; width: 216px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image192.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 91px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image193.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 24px; width: 39px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image194.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 13px; width: 15px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image034.png\" \/>\u00a0Length of shortest distance = <img decoding=\"async\" style=\"height: 103px; width: 273px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image195.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 44px; width: 152px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image196.png\" \/>\u00a0(numerically)<\/p>\n<p>= <img decoding=\"async\" style=\"height: 24px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image197.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112-6\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"16Find_the_shortest_distance_between_the_lines_whose_vector_equations_are\"><\/span><strong>16.Find the shortest distance between the lines whose vector equations are<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong><img decoding=\"async\" style=\"height: 35px; width: 197px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image198.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 35px; width: 219px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image199.png\" \/><\/strong><\/p>\n<p><strong>Ans. <\/strong>Equation of the first line is <img decoding=\"async\" style=\"height: 35px; width: 209px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image200.png\" \/><\/p>\n<p>Comparing this equation with <img decoding=\"async\" style=\"height: 27px; width: 77px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image171.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 99px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image201.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 97px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image202.png\" \/><\/p>\n<p>Again equation of second line <img decoding=\"async\" style=\"height: 35px; width: 219px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image203.png\" \/><\/p>\n<p>Comparing this equation with <img decoding=\"async\" style=\"height: 27px; width: 81px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image204.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 109px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image205.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 28px; width: 105px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image206.png\" \/><\/p>\n<p>Now shortest distance <img decoding=\"async\" style=\"height: 27px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image207.png\" \/>\u00a0=\u00a0 <img decoding=\"async\" style=\"height: 53px; width: 101px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image208.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p>Here<img decoding=\"async\" style=\"height: 35px; width: 327px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image209.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 70px; width: 365px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image210.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 36px; width: 291px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image211.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 29px; width: 365px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image212.png\" \/><\/p>\n<p>Putting these values in eq. (i),<\/p>\n<p>Shortest distance <img decoding=\"async\" style=\"height: 47px; width: 176px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image213.png\" \/><\/p>\n<hr \/>\n<h4 style=\"text-align: center;\"><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112-7\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<h4><span class=\"ez-toc-section\" id=\"17Find_the_shortest_distance_between_the_lines_whose_vector_equations_are\"><\/span><strong>17.Find the shortest distance between the lines whose vector equations are <\/strong><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><strong><img decoding=\"async\" style=\"height: 29px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image214.png\" \/>\u00a0and <img decoding=\"async\" style=\"height: 29px; width: 219px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image215.png\" \/>\u00a0<\/strong><\/p>\n<p><strong>Ans. <\/strong>Equation of first line is <img decoding=\"async\" style=\"height: 29px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image214.png\" \/><\/p>\n<p>=\u00a0 <img decoding=\"async\" style=\"height: 27px; width: 159px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image216.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 183px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image217.png\" \/><\/p>\n<p>Comparing this equation with <img decoding=\"async\" style=\"height: 27px; width: 48px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image218.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 220px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image219.png\" \/><\/p>\n<p>Equation of second line is <img decoding=\"async\" style=\"height: 29px; width: 216px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image220.png\" \/><\/p>\n<p>= <img decoding=\"async\" style=\"height: 27px; width: 156px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image221.png\" \/>\u00a0= <img decoding=\"async\" style=\"height: 35px; width: 168px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image222.png\" \/><\/p>\n<p>Comparing this equation with <img decoding=\"async\" style=\"height: 27px; width: 53px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image223.png\" \/>,<\/p>\n<p><img decoding=\"async\" style=\"height: 28px; width: 207px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image224.png\" \/><\/p>\n<p>Now Shortest distance <img decoding=\"async\" style=\"height: 27px; width: 27px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image207.png\" \/>\u00a0=\u00a0 <img decoding=\"async\" style=\"height: 64px; width: 121px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image225.png\" \/>\u2026\u2026\u2026.(i)<\/p>\n<p>Here<img decoding=\"async\" style=\"height: 35px; width: 266px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image226.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 72px; width: 395px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image227.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 33px; width: 228px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image228.png\" \/><\/p>\n<p><img decoding=\"async\" style=\"height: 32px; width: 424px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image229.png\" \/><\/p>\n<p>Putting these values in eq. (i),<\/p>\n<p>Shortest distance <img decoding=\"async\" style=\"height: 41px; width: 106px;\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/ncert\/12\/maths\/Ex11.2\/image230.png\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"NCERT_Solutions_class_12_Maths_Exercise_112-8\"><\/span>NCERT Solutions class 12 Maths Exercise 11.2<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 11.2 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 11.2\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/ncert-solutions-class-12-maths-exercise-11-2\/\" aria-label=\"More on NCERT Solutions class 12 Maths Exercise 11.2\">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-4536","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 11.2 | myCBSEguide<\/title>\n<meta name=\"description\" content=\"NCERT Solutions class 12 Maths Exercise 11.2 in PDF format for free download. 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