{"id":32454,"date":"2022-03-29T17:00:42","date_gmt":"2022-03-29T11:30:42","guid":{"rendered":"https:\/\/mcq-questions.com\/?p=32454"},"modified":"2022-03-29T17:25:33","modified_gmt":"2022-03-29T11:55:33","slug":"ncert-solutions-for-class-12-maths-chapter-10-ex-10-3","status":"publish","type":"post","link":"https:\/\/mcq-questions.com\/ncert-solutions-for-class-12-maths-chapter-10-ex-10-3\/","title":{"rendered":"NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3"},"content":{"rendered":"

These NCERT Solutions for Class 12 Maths<\/a> Chapter 10 Vector Algebra Ex 10.3 Questions and Answers are prepared by our highly skilled subject experts. https:\/\/mcq-questions.com\/ncert-solutions-for-class-12-maths-chapter-10-ex-10-3\/<\/p>\n

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3<\/h2>\n

\"NCERT<\/p>\n

Class 12 Maths Ncert Solutions Chapter 10.3 Question 1.<\/strong>
\nFind the angle between two vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) with magnitudes \\(\\sqrt{3}\\) and 2 respectively, and such that \\(\\vec{a}\\).\\(\\vec{b}\\) = \\(\\sqrt{6}\\)
\nSolution:
\n|\\(\\vec{a}\\)|= \\(\\sqrt{3}\\)
\nand \\(\\vec{b}\\) = 2, \\(\\vec{a}\\).\\(\\vec{b}\\) = \\(\\sqrt{6}\\)
\nLet \u03b8 be the angle between \\(\\vec{a}\\) and \\(\\vec{b}\\). Then
\n\\(\\cos \\theta=\\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}||\\vec{b}|}=\\frac{\\sqrt{6}}{(\\sqrt{3})(2)}=\\frac{1}{\\sqrt{2}}\\)
\n\u2234 \u03b8 = \\(\\cos ^{-1}\\left(\\frac{1}{\\sqrt{2}}\\right)=\\frac{\\pi}{4}\\)<\/p>\n

Question 2.
\nFind the angle between the vectors \\(\\hat{i}-2 \\hat{j}+3 \\hat{k} \\text { and } 3 \\hat{i}-2 \\hat{j}+\\hat{k}\\)
\nSolution:
\nLet \\(\\vec{a}\\) = \\(\\hat{i}-2 \\hat{j}+3 \\hat{k}\\), \\(\\vec{b}\\) = \\(\\hat{3i}-2 \\hat{j}+ \\hat{k}\\)
\n\\(\\vec{a}\\).\\(\\vec{b}\\) = \\((\\hat{i}-2 \\hat{j}+3 \\hat{k}) \\cdot(3 \\hat{i}-2 \\hat{j}+\\hat{k})\\)
\n= 1(3) + (- 2)(- 2) + 3(1) = 3 + 4 + 3 = 10
\n\"NCERT<\/p>\n

Question 3.
\nFind the projection of the vector \\(\\overrightarrow { i } -\\overrightarrow { j }\\), on the line represented by the vector \\(\\overrightarrow { i } +\\overrightarrow { j }\\).
\nSolution:
\n\"NCERT<\/p>\n

Question 4.
\nFind the projection of the vector \\(\\hat{i}+3 \\hat{j}+7 \\hat{k}\\) on the vector \\(7 \\hat{i}-\\hat{j}+8 \\hat{k}\\)
\nSolution:
\n\"NCERT<\/p>\n

\"NCERT<\/p>\n

Question 5.
\nShow that each of the given three vectors is a unit vector \\(\\frac { 1 }{ 7 } \\left( 2\\hat { i } +3\\hat { j } +6\\hat { k } \\right) ,\\frac { 1 }{ 7 } \\left( 3\\hat { i } -6\\hat { j } +2\\hat { k } \\right) ,\\frac { 1 }{ 7 } \\left( 6\\hat { i } +2\\hat { j } -3\\hat { k } \\right)\\)
\nAlso show that they are mutually perpendicular to each other.
\nSolution:
\nLet \\(\\vec{a}\\) = \\(\\frac{1}{7}(2 \\hat{i}+3 \\hat{j}+6 \\hat{k})\\), \\(\\vec{b}\\) = \\(\\frac{1}{7}(3 \\hat{i}-6 \\hat{j}+2 \\hat{k})\\) and \\(\\vec{c}\\) = \\(\\frac{1}{7}(6 \\hat{i}+2 \\hat{j}-3 \\hat{k})\\)
\n\"NCERT
\nHere \\(\\vec{a}\\).\\(\\vec{b}\\) = 0, \\(\\vec{b}\\).\\(\\vec{c}\\) = 0 and \\(\\vec{a}\\).\\(\\vec{c}\\) = 0
\n\u2234 The vectors \\(\\vec{a}\\).\\(\\vec{b}\\) and \\(\\vec{c}\\) are mutually per-pendicular vectors.<\/p>\n

Question 6.
\n\\(Find\\left| \\overrightarrow { a } \\right| and\\left| \\overrightarrow { b } \\right| if\\left( \\overrightarrow { a } +\\overrightarrow { b } \\right) \\cdot \\left( \\overrightarrow { a } -\\overrightarrow { b } \\right) =8\\quad and\\left| \\overrightarrow { a } \\right| =8\\left| \\overrightarrow { b } \\right| \\)
\nSolution:
\n\"NCERT<\/p>\n

\"NCERT<\/p>\n

Question 7.
\nEvaluate the product :
\n\\((3 \\vec{a}-5 \\vec{b}) \\cdot(2 \\vec{a}+7 \\vec{b})\\)
\nSolution:
\n\\(\\left( 3\\overrightarrow { a } -5\\overrightarrow { b } \\right) \\cdot \\left( 2\\overrightarrow { a } +7\\overrightarrow { b } \\right) \\)
\n\\(=6\\overrightarrow { a } .\\overrightarrow { a } -10\\overrightarrow { b } \\overrightarrow { a } +21\\overrightarrow { a } .\\overrightarrow { b } -35\\overrightarrow { b } .\\overrightarrow { b } \\)
\n\\(=6{ \\left| \\overrightarrow { a } \\right| }^{ 2 }-11\\overrightarrow { a } \\overrightarrow { b } -35{ \\left| \\overrightarrow { b } \\right| }^{ 2 }\\)<\/p>\n

Question 8.
\nFind the magnitude of two vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) having the same magnitude and such that the angle between them is 60\u00b0 and their scalar product is \\(\\frac { 1 }{ 2 }\\)
\nSolution:
\n\"NCERT<\/p>\n

Question 9.
\nFind |\\(\\vec{a}\\)| , if for a unit vector \\(\\vec{a}\\), \\((\\vec{x}-\\vec{a}) \\cdot(\\vec{x}+\\vec{a})\\) = 12
\nSolution:
\n\"NCERT<\/p>\n

Question 10.
\nIf \\(\\overrightarrow { a } =2\\hat { i } +2\\hat { j } +3\\hat { k } ,\\overrightarrow { b } =-\\hat { i } +2\\hat { j } +\\hat { k } and \\overrightarrow { c } =3\\hat { i } +\\hat { j } \\) such that \\(\\overrightarrow { a } +\\lambda \\overrightarrow { b } \\bot \\overrightarrow { c } \\) , then find the value of \u03bb.
\nSolution:
\n\"NCERT<\/p>\n

Question 11.
\nShow that \\(|\\vec{a}| \\vec{b}+|\\vec{b}| \\vec{a} \\), is per-pendicular to \\(|\\vec{a}| \\vec{b}-|\\vec{b}| \\vec{a} \\) for any two non-zero vectors \\(\\vec{a} \\text { and } \\vec{b}\\)
\nSolution:
\n\"NCERT<\/p>\n

\"NCERT<\/p>\n

Question 12.
\nIf \\(\\overrightarrow { a } \\cdot \\overrightarrow { a } =0\\quad and\\quad \\overrightarrow { a } \\cdot \\overrightarrow { b } =0\\), then what can be concluded about the vector \\(\\overrightarrow { b } \\) ?
\nSolution:
\n\\(\\vec{a}\\).\\(\\vec{a}\\) = 0 \u21d2 \\(\\vec{a}\\) is a zero vector.
\nsince \\(\\vec{a}\\) = \\(\\vec{0}\\), \\(\\vec{a}\\).\\(\\vec{b}\\) = 0 for any vector \\(\\vec{b}\\)
\nVector \\(\\vec{b}\\) be any vector.<\/p>\n

Question 13.
\nIf \\(\\overrightarrow { a } ,\\overrightarrow { b } ,\\overrightarrow { c } \\) are the unit vector such that \\(\\overrightarrow { a } +\\overrightarrow { b } +\\overrightarrow { c } =0\\) , then find the value of \\(\\overrightarrow { a } .\\overrightarrow { b } +\\overrightarrow { b } .\\overrightarrow { c } +\\overrightarrow { c } .\\overrightarrow { a } \\)
\nSolution:
\n\"NCERT<\/p>\n

Question 14.
\nIf either vector \\(\\vec{a}=\\overrightarrow{0} \\text { or } \\vec{b}=\\overrightarrow{0}\\), then \\(\\vec{a} \\cdot \\vec{b}\\). But the converse need not be true. Justify your answer with an example.
\nSolution:
\n\"NCERT
\nThus two non-zero vectors \\(\\vec{a}\\) and \\(\\vec{b}\\) may have \\(\\vec{a}\\).\\(\\vec{a}\\) zero.<\/p>\n

Question 15.
\nIf the vertices A,B,C of a triangle ABC are (1, 2, 3) (-1, 0, 0), (0, 1, 2) respectively, then find \u2220ABC.
\nSolution:
\n\"NCERT<\/p>\n

Question 16.
\nShow that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, -1) are collinear.
\nSolution:
\nA (1, 2, 7), B (2, 6, 3) and C (3, 10, -1) are the points.
\n\"NCERT
\n\u2234 \\(\\overrightarrow{AB}\\)\\(\\overrightarrow{AC}\\) are parallel and A is a common point. Therefore A, B, C are collinear.<\/p>\n

\"NCERT<\/p>\n

Question 17.
\nShow that the vectors \\(2 \\hat{i}-\\hat{j}+\\hat{k}, \\hat{i}-3 \\hat{j}-5 \\hat{k} \\text { and } \\quad 3 \\hat{i}-4 \\hat{j}-4 \\hat{k}\\) from the vertices of a right angled triangle.
\nSolution:
\nLet A, B and C be the vertices of the triangle with the vectors \\(2 \\hat{i}-\\hat{j}+\\hat{k}, \\hat{i}-3 \\hat{j}-5 \\hat{k} \\text { and } \\quad 3 \\hat{i}-4 \\hat{j}-4 \\hat{k}\\)
\n\u2234 \\(\\overrightarrow{AB}\\) = p.v. of B – p.v. of A
\n\"NCERT
\n\u2234 A, B, C are the vertices of \u2206 ABC
\n\\(\\overrightarrow{BC}\\).\\(\\overrightarrow{CA}\\) = (2)(-1) + (-1)(3) + (1)(5)
\n= – 2 – 3 + 5 = 0
\n\u2234 \\(\\overrightarrow{BC}\\)\u22a5\\(\\overrightarrow{CA}\\)
\nHence triangle ABC is a right triangle<\/p>\n

Question 18.
\nIf \\(\\overrightarrow { a } \\) is a non-zero vector of magnitude \u2018a\u2019 and \u03bb is a non- zero scalar, then \u03bb \\(\\overrightarrow { a } \\) is unit vector if
\n(a) \u03bb = 1
\n(b) \u03bb = – 1
\n(c) a = |\u03bb|
\n(d) a = \\(\\frac { 1 }{ \\left| \\lambda \\right| } \\)
\nSolution:
\n\\(\\left| \\overrightarrow { a } \\right| =a\\)
\nGiven : \\(\\lambda \\overrightarrow { a } \\) is a unit vectors.
\n\\(|\\lambda \\vec{a}|=1 \\Rightarrow|\\lambda||\\vec{a}|=1 \\Rightarrow|\\lambda| a=1 \\Rightarrow a=\\frac{1}{|\\lambda|}\\)<\/p>\n","protected":false},"excerpt":{"rendered":"

These NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Questions and Answers are prepared by our highly skilled subject experts. https:\/\/mcq-questions.com\/ncert-solutions-for-class-12-maths-chapter-10-ex-10-3\/ NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3 Class 12 Maths Ncert Solutions Chapter 10.3 Question 1. Find the angle between two vectors and with magnitudes …<\/p>\n

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3<\/span> Read More »<\/a><\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"default","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","spay_email":""},"categories":[3],"tags":[],"yoast_head":"\nNCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 - MCQ Questions<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mcq-questions.com\/ncert-solutions-for-class-12-maths-chapter-10-ex-10-3\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 - MCQ Questions\" \/>\n<meta property=\"og:description\" content=\"These NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Questions and Answers are prepared by our highly skilled subject experts. https:\/\/mcq-questions.com\/ncert-solutions-for-class-12-maths-chapter-10-ex-10-3\/ NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3 Class 12 Maths Ncert Solutions Chapter 10.3 Question 1. 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