{"id":4900,"date":"2023-12-27T12:20:00","date_gmt":"2023-12-27T06:50:00","guid":{"rendered":"https:\/\/wordpress-505192-1602719.cloudwaysapps.com\/?p=4900"},"modified":"2023-12-28T17:47:27","modified_gmt":"2023-12-28T12:17:27","slug":"samacheer-kalvi-12th-maths-solutions-chapter-6-ex-6-10","status":"publish","type":"post","link":"https:\/\/samacheerkalviguru.com\/samacheer-kalvi-12th-maths-solutions-chapter-6-ex-6-10\/","title":{"rendered":"Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10"},"content":{"rendered":"

You can Download Samacheer Kalvi 12th Maths Book Solutions<\/a> Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.<\/p>\n

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10<\/h2>\n

Choose the correct or the most suitable answer from the given four alternatives:
\nQuestion 1.
\nIf \\(\\vec{a}\\) and \\(\\vec{b}\\) are parallel vector, then \\([\\vec{a}, \\vec{c}, \\vec{b}]\\) is equal to ………………..
\n(a) 2
\n(b) -1
\n(c) 1
\n(d) 0
\nSolution:
\n(d) 0
\nHint:
\n\\(\\vec{a}\\) and \\(\\vec{b}\\) are parallel vectors, so \\(\\vec{a} \\times \\vec{b}\\) = 0
\nthen \\([\\vec{a} \\vec{c} \\vec{b}]=-[\\vec{a} \\vec{b} \\vec{c}]=-(\\vec{a} \\times \\vec{b}) \\cdot \\vec{c}\\) = 0<\/p>\n

Question 2.
\nIf a vector \\(\\vec{\\alpha}\\) lies in the plane of \\(\\vec{\\beta}\\) and \\(\\vec{\\gamma}\\), then …………
\n\"Samacheer
\nSolution:
\n(c) \\([\\vec{\\alpha}, \\vec{\\beta}, \\vec{\\gamma}]\\) = 0
\nHint:
\nSince \\([\\vec{\\alpha}, \\vec{\\beta}, \\vec{\\gamma}]\\) are lie in the same plane
\nso \\([\\vec{\\alpha}, \\vec{\\beta}, \\vec{\\gamma}]\\) = 0<\/p>\n

Question 3.
\nIf \\(\\vec{a} \\cdot \\vec{b}=\\vec{b} \\cdot \\vec{c}=\\vec{c} \\cdot \\vec{a}\\)= 0, then the value of \\(|[\\vec{a}, \\vec{b}, \\vec{c}]|\\) is ……………….
\n(a) \\(|\\vec{a}||\\vec{b}||\\vec{c}|\\)
\n(b) \\(\\frac{1}{3}|\\vec{a}||\\vec{b}||\\vec{c}|\\)
\n(c) 1
\n(d) -1
\nSolution:
\n(a) \\(|\\vec{a}||\\vec{b}||\\vec{c}|\\)
\nHint:
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 4.
\nIf \\(\\vec{a}, \\vec{b}, \\vec{c}\\) are three unit vectors such that \\(\\vec{a}\\) is perpendicular to \\(\\vec{b}\\), and is parallel to \\(\\vec{c}\\) then \\(\\vec{a} \\times(\\vec{b} \\times \\vec{c})\\)is equal to ………………….
\n(a) \\(\\vec{a}\\)
\n(b) \\(\\vec{b}\\)
\n(c) \\(\\vec{c}\\)
\n(d) \\(\\vec{0}\\)
\nSolution:
\n(b) \\(\\vec{b}\\)
\nHint:
\n\"Samacheer<\/p>\n

Question 5.
\n\"Samacheer
\n(a) 1
\n(b) -1
\n(c) 2
\n(d) 3
\nSolution:
\n(a) 1
\nHint:
\n\"Samacheer<\/p>\n

Question 6.
\nThe volume of the parallelepiped with its edges represented by the vectors \\(\\hat{i}+\\hat{j}, i+2 \\hat{j}\\), \\(\\hat{i}+\\hat{j}+\\pi \\hat{k}\\) is ……………
\n\"Samacheer
\nSolution:
\n(c) \u03c0
\nHint:
\nVolume = \\([\\vec{a} \\vec{b} \\vec{c}]=\\left|\\begin{array}{ccc}{1} & {1} & {0} \\\\ {1} & {2} & {0} \\\\ {1} & {1} & {\\pi}\\end{array}\\right|\\)
\n= \u03c0(2 – 1) = \u03c0 cubic units<\/p>\n

\"Samacheer<\/p>\n

Question 7.
\nIf \\(\\vec{a}\\) and \\(\\vec{b}\\) are unit vectors such that \\([\\vec{a}, \\vec{b}, \\vec{a} \\times \\vec{b}]=\\frac{\\pi}{4}\\) then the angle between \\(\\vec{a}\\) and \\(\\vec{b}\\) is …………
\n\"Samacheer
\nSolution:
\n(a) \\(\\frac{\\pi}{6}\\)
\nHint:
\n\"Samacheer<\/p>\n

Question 8.
\n\"Samacheer
\n(a) 0
\n(b) 1
\n(c) 6
\n(d) 3
\nSolution:
\n(a) 0
\nHint:
\n\"Samacheer
\nEquate corresponding coefficients on both sides
\n\u03bb + \u00b5 = 0 and \u03bb = -1 this gives \u00b5 = 1
\n\u2234 Then the value of \u03bb + \u00b5 = 0.<\/p>\n

Question 9.
\nIf \\(\\vec{a}, \\vec{b}, \\vec{c}\\) are three non-coplanar vectors such that \\(\\vec{a} \\times(\\vec{b} \\times \\vec{c})=\\frac{\\vec{b}+\\vec{c}}{\\sqrt{2}}\\), then the angle between \\(\\vec{a}\\) and \\(\\vec{b}\\) is …………
\n(a) 81
\n(b) 9
\n(c) 27
\n(d) 18
\nSolution:
\n(a) 81
\nHint:
\n\"Samacheer<\/p>\n

Question 10.
\n\"Samacheer
\nSolution:
\n(b) \\(\\frac{3 \\pi}{4}\\)
\nHint:
\n\"Samacheer<\/p>\n

Question 11.
\nIf the volume of the parallelepiped with \\(\\vec{a} \\times \\vec{b}, \\vec{b} \\times \\vec{c}, \\vec{c} \\times \\vec{a}\\) as coterminous edges is 8 cubic units, then the volume of the parallelepiped with \\((\\vec{a} \\times \\vec{b}) \\times(\\vec{b} \\times \\vec{c}),(\\vec{b} \\times \\vec{c}) \\times(\\vec{c} \\times \\vec{a})\\) and \\((\\vec{c} \\times \\vec{a}) \\times(\\vec{a} \\times \\vec{b})\\) as coterminous edges is ………………
\n(a) 8 cubic units
\n(b) 512 cubic units
\n(c) 64 cubic units
\n(d) 24 cubic units
\nSolution:
\n(c) 64 cubic units
\nHint:
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 12.
\nConsider the vectors \\(\\vec{a}, \\vec{b}, \\vec{c}, \\vec{d}\\) such that \\((\\vec{a} \\times \\vec{b}) \\times(\\vec{c} \\times \\vec{d})=\\overrightarrow{0}\\). Let P1<\/sub> and P2<\/sub> be the planes determined by the pairs of vectors \\(\\vec{a}, \\vec{b}\\) and \\(\\vec{c}, \\vec{d}\\) respectively. Then the angle between P1<\/sub> and P2<\/sub> is ……………..
\n(a) 0\u00b0
\n(b) 45\u00b0
\n(c) 60\u00b0
\n(d) 90\u00b0
\nSolution:
\n(a) 0\u00b0
\nHint:
\n\"Samacheer<\/p>\n

Question 13.
\n\"Samacheer
\n(a) perpendicular
\n(b) parallel
\n(c) inclined at an angle \\(\\frac{\\pi}{3}\\)
\n(d) inclined at an angle \\(\\frac{\\pi}{6}\\)
\nSolution:
\n(b) parallel
\nHint:
\n\"Samacheer<\/p>\n

Question 14.
\nIf \"Samacheer, then a vector perpendicular to \\(\\vec{a}\\) and lies in the plane containing \\(\\vec{b}\\) and \\(\\vec{c}\\) is
\n\"Samacheer
\nSolution:
\n(d) \\(-17 \\hat{i}-21 \\hat{j}-97 \\hat{k}\\)
\nHint:
\n\"Samacheer<\/p>\n

Question 15.
\nThe angle between the lines \"Samacheer is ………..
\n\"Samacheer
\nSolution:
\n(d) \\(\\frac{\\pi}{2}\\)
\nHint:
\n\"Samacheer
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 16.
\nIf the line \"Samacheer lies in the plane x + 3 + \u03b1z + \u03b2 = 0, then (\u03b1, \u03b2) is …………..
\n(a) (-5, 5)
\n(b) (-6, 7)
\n(c) (5, -5)
\n(d) (6, -7)
\nSolution:
\n(d) (-6, 7)
\nHint:
\nd.c.s of the first line = (3, -5, 2)
\nd.c.s of the line perpendicular to plane = (1, 3, -\u03b1)
\na1<\/sub>a2<\/sub> + b1<\/sub>b2<\/sub> + c1<\/sub>c2<\/sub> = 0
\n3 – 15 – 2\u03b1 = 0 => -12 – 2\u03b1 = 0
\n-2\u03b1 =12 => \u03b1 = -6
\nPlane passes through the point (2, 1, -2) so
\n2 + 3 + 6(-2) + \u03b2 = 0 => \u03b2 = 7
\n(\u03b1, \u03b2) = (-6, 7)<\/p>\n

Question 17.
\nThe angle between the line \\(\\vec{r}=(\\hat{i}+2 \\hat{j}-3 \\hat{k})+t(2 \\hat{i}+\\hat{j}-2 \\hat{k})\\) and the plane \\(\\vec{r} \\cdot(\\hat{i}+\\hat{j})+4\\) = 0 is ……………
\n(a) 0\u00b0
\n(b) 30\u00b0
\n(c) 45\u00b0
\n(d) 90\u00b0
\nSolution:
\n(c) 45\u00b0
\nHint:
\n\"Samacheer<\/p>\n

Question 18.
\nThe coordinates of the point where the line \\(\\vec{r}=(6 \\hat{i}-\\hat{j}-3 \\hat{k})+t(-\\hat{i}+4 \\hat{k})\\) meets the plane \\(\\vec{r} \\cdot(\\hat{i}+\\hat{j}-\\hat{k})\\) are
\n(a) (2, 1, 0)
\n(b) (7, -1, -7)
\n(c) (1, 2, -6)
\n(c) (5, -1, 1)
\nSolution:
\n(d) (5, -1, 1)
\nHint:
\nCartesian equation of the line
\n\\(\\frac{x-6}{-1}=\\frac{y+1}{0}+\\frac{z+3}{4}\\) = \u03bb
\n(-\u03bb + 6, -1, 4\u03bb – 3)
\nThis meets the plane x + y – z = 3
\n-\u03bb + 6 – 1 – 41 + 3 = 3 \u21d2 -5\u03bb = -5
\n\u03bb = 1
\nThe required point (5, -1, 1).<\/p>\n

Question 19.
\nDistance from the origin to the plane 3x – 6y + 2z + 7 = 0 is ………………
\n(a) 0
\n(b) 1
\n(c) 2
\n(d) 3
\nSolution:
\n(b) 1
\nHint:
\nDistance from the origin (0, 0, 0) to the plane
\n\"Samacheer<\/p>\n

Question 20.
\nThe distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is ……………
\n\"Samacheer
\nSolution:
\n(a) \\(\\frac{\\sqrt{7}}{2 \\sqrt{2}}\\)
\nHint:
\nx + 2y + 3z + 1 = 0; 2x + 4y + 6z + 7 = 0
\nMultiplying 2 on both sides
\n2x + 4y + 6z + 14 = 0 .
\na = 2, b = 4, c = 6, d1<\/sub> = 14, d2<\/sub> = ?
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 21.
\nIf the direction cosines of a line are \\(\\frac{1}{c}, \\frac{1}{c}, \\frac{1}{c}\\), then ……………….
\n(a) c = \u00b13
\n(b) c = \u00b1 \\(\\sqrt{3}\\)
\n(c) c > 0
\n(d) 0 < c < 1
\nSolution:
\n(b) c = \u00b1 \\(\\sqrt{3}\\)
\nHint:
\nWe know that sum of the squares of direction cosines = 1
\n\"Samacheer<\/p>\n

Question 22.
\nThe vector equation \\(\\vec{r}=(\\hat{i}-2 \\hat{j}-\\hat{k})+t(6 \\vec{j}-\\hat{k})\\) represents a straight line passing through the points ……………. (a) (0, 6, -1) and (1, -2, -1) (b) (0, 6, -1) and (-1, -4, -2) (c) (1, -2, -1) and (1, 4, -2) (d) (1, -2, -1) and (0, -6, 1)
\nSolution:
\n(c) (1, -2, -1) and (1, 4, -2)
\nHint:
\nThe required vector equation is \\(\\vec{r}=\\vec{a}+t(\\vec{b}-\\vec{a})\\)
\n\"Samacheer
\nFrom (1) and (2) The points are (1, -2, -1) and (1, 4, -2)<\/p>\n

Question 23.
\nIf the distance of the point (1, 1, 1) from the origin is half of its distance from the plane x + y + z + k= 0, then the values of k are ………….. (a) \u00b13 (b) \u00b16 (c) -3, 9 (d) 3, -9
\nSolution:
\n(d) 3, -9
\nHint:
\n\"Samacheer<\/p>\n

Question 24.
\nIf the planes \\(\\vec{r} \\cdot(2 \\hat{i}-\\lambda \\hat{j}+\\hat{k})=3 \\text { and } \\vec{r} \\cdot(4 \\hat{i}+\\hat{j}-\\mu \\hat{k})\\) = 5 are parallel, then the value of \u03bb and \u00b5 are ……………
\nSolution:
\n(c) \\(-\\frac{1}{2}\\), -2
\nHint:
\n\"Samacheer<\/p>\n

Question 25.
\nIf the length of the perpendicular from the origin to the plane 2x + 3y + \u03bbz = 1, \u03bb > 0 is \\(-\\frac{1}{5}\\), then the value of \u03bb is …………..
\n(a) \\(2 \\sqrt{3}\\)
\n(b) \\(3 \\sqrt{2}\\)
\n(c) 0
\n(d) 1
\nSolution:
\n(a) \\(2 \\sqrt{3}\\)
\nHint:
\nGiven length of perpendicular from origin to the plane = \\(-\\frac{1}{5}\\)
\n\"Samacheer<\/p>\n

Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 Additional Problems<\/h3>\n

Question 1.
\n\"Samacheer
\n(a) 6
\n(b) 10
\n(c) 12
\n(d) 24
\nSolution:
\n(c) 12
\nHint:
\n\"Samacheer<\/p>\n

Question 2.
\n\"Samacheer
\n(a) only x
\n(b) only y
\n(c) Neither x nor y
\n(d) Both x and y
\nSolution:
\n(c) Neither x nor y
\nHint:
\n\"Samacheer<\/p>\n

Question 3.
\n\"Samacheer
\n(a) 0
\n(b) 1
\n(c) 2
\n(d) 3
\nSolution:
\n(b) 3
\nHint:
\n\"Samacheer<\/p>\n

Question 4.
\nThe value of \"Samacheer = ………………
\n(a) 1
\n(b) 3
\n(c) -3
\n(d) 0
\nSolution:
\n(b) 3
\nHint:
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 5.
\nLet a, b, c be distinct non-negative numbers. If the vectors \"Samacheer lie in a plane, then c is ……………..
\n(a) the A.M. of a and b
\n(b) the G.M. of a and b
\n(c) the H.M. of a and b
\n(d) equal to zero.
\nSolution:
\n(b) the G.M. of a and b
\nHint:
\n\"Samacheer<\/p>\n

Question 6.
\nThe value of \\(\\hat{i} \\cdot(\\hat{j} \\times \\hat{k})+(\\hat{i} \\times \\hat{k}) \\cdot \\hat{j}\\) ………….
\n(a) 1
\n(b) -1
\n(c) 0
\n(d) \\(\\hat{j}\\)
\nSolution:
\n(c) 0
\nHint:
\n\"Samacheer<\/p>\n

Question 7.
\nThe value of \\((\\hat{i}-\\hat{j}, \\hat{j}-\\hat{k}, \\hat{k}-\\hat{i})\\) is …………..
\n(a) 0
\n(b) 1
\n(c) 2
\n(d) 3
\nSolution:
\n(a) 0
\nHint:
\n\"Samacheer
\n\"Samacheer<\/p>\n

Question 8.
\n\"Samacheer
\nSolution:
\n(c) \\(\\vec{u}=\\overrightarrow{0}\\)
\nHint:
\n\"Samacheer<\/p>\n

Question 9.
\nThe area of the parallelogram having a diagonal \\(3 \\vec{i}+\\vec{j}-\\vec{k}\\) and a side \\(\\vec{i}-3 \\vec{j}+4 \\vec{k}\\) is ………………
\n\"Samacheer
\nSolution:
\n(d) \\(3 \\sqrt{30}\\)
\nSolution:
\n\"Samacheer<\/p>\n

\"Samacheer<\/p>\n

Question 10.
\nIf \"Samacheer, then ……………….
\n\"Samacheer
\nSolution:
\n(d) \\(\\vec{x}=\\overrightarrow{0} \\text { or } \\vec{y}=\\overrightarrow{0} \\text { or } \\vec{x} \\text { and } \\vec{y}\\) are parallel
\nHint:
\n\"Samacheer<\/p>\n

Question 11.
\nIf \\(\\overrightarrow{\\mathrm{PR}}=2 \\vec{i}+\\vec{j}+\\vec{k}, \\overrightarrow{\\mathrm{Question}}=-\\vec{i}+3 \\vec{j}+2 \\vec{k}\\), then the area of the quadrilateral PQRS is ………………..
\n\"Samacheer
\nSolution:
\n(c) \\(\\frac{5 \\sqrt{3}}{2}\\)
\nHint:
\n\"Samacheer<\/p>\n

Question 12.
\n\"Samacheer
\nSolution:
\n(c) \\(\\vec{c}\\) parallel to \\(\\vec{a}\\)
\nHint:
\n\"Samacheer<\/p>\n","protected":false},"excerpt":{"rendered":"

You can Download Samacheer Kalvi 12th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations. Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 Choose the correct or the most suitable answer from the given four alternatives: …<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[8],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts\/4900"}],"collection":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/comments?post=4900"}],"version-history":[{"count":1,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts\/4900\/revisions"}],"predecessor-version":[{"id":60225,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/posts\/4900\/revisions\/60225"}],"wp:attachment":[{"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/media?parent=4900"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/categories?post=4900"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/samacheerkalviguru.com\/wp-json\/wp\/v2\/tags?post=4900"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}