Tamil Nadu Board of Secondary EducationHSC Science Class 12th
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Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 6 - Applications of Vector Algebra [Latest edition]

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Class 12th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com
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Chapter 6: Applications of Vector Algebra

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6Exercise 6.7Exercise 6.8Exercise 6.9Exercise 6.10
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Exercise 6.1 [Page 231]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.1 [Page 231]

Exercise 6.1 | Q 1 | Page 231

Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord

Exercise 6.1 | Q 2 | Page 231

Prove by vector method that the median to the base of an isosceles triangle is perpendicular to the base

Exercise 6.1 | Q 3 | Page 231

Prove by vector method that an angle in a semi-circle is a right angle

Exercise 6.1 | Q 4 | Page 231

Prove by vector method that the diagonals of a rhombus bisect each other at right angles

Exercise 6.1 | Q 5 | Page 231

Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle

Exercise 6.1 | Q 6 | Page 231

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is `1/2 |bar"AC" xx bar"BD"|`

Exercise 6.1 | Q 7 | Page 231

Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area

Exercise 6.1 | Q 8 | Page 231

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = `1/3` (area of ΔABC)

Exercise 6.1 | Q 9 | Page 231

Using vector method, prove that cos(α – β) = cos α cos β + sin α sin β

Exercise 6.1 | Q 10 | Page 231

Prove by vector method that sin(α + ß) = sin α cos ß + cos α sin ß

Exercise 6.1 | Q 11 | Page 231

A particle acted on by constant forces `8hat"i" + 2hat"j" - 6hat"k"` and `6hat"i" + 2hat"j" - 2hat"k"` is displaced from the point (1, 2, 3) to the point (5, 4, 1). Find the total work done by the forces

Exercise 6.1 | Q 12 | Page 231

Forces of magnitudes `5sqrt(2)` and `10sqrt(2)` units acting in the directions `3hat"i" + 4hat"j" + 5hat"k"` and `10hat"i" + 6hat"j" - 8hat"k"` respectively, act on a particle which is displaced from the point with position vector `4hat"i" - 3hat"j" - 2hat"k"` to the point with position vector `6hat"i" + hat"j" - 3hat"k"`. Find the work done by the forces

Exercise 6.1 | Q 13 | Page 231

Find the magnitude and direction cosines of the torque of a force represented by `3hat"i" + 4hat"j" - 5hat"k"` about the point with position vector `2hat"i" - 3hat"j" + 4hat"k"`  acting through a point whose position vector is `4hat"i" + 2hat"j" - 3hat"k"`

Exercise 6.1 | Q 14 | Page 231

Find the torque of the resultant of the three forces represented by `- 3hat"i" + 6hat"j" - 3hat"k", 4hat"i" - 10hat"j" + 12hat"k"` and `4hat"i" + 7hat"j"` acting at the point with position vector `8hat"i" - 6hat"j" - 4hat"k"` about the point with position vector `18hat"i" + 3hat"j" - 9hat"k"`

Exercise 6.2 [Pages 237 - 238]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.2 [Pages 237 - 238]

Exercise 6.2 | Q 1 | Page 237

If `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" * (vec"b" xx vec"c")`

Exercise 6.2 | Q 2 | Page 237

Find the volume of the parallelepiped whose coterminous edges are represented by the vectors `- 6hat"i" + 14hat"j" + 10hat"k", 14hat"i" - 10hat"j" - 6hat"k"` and `2hat"i" + 4hat"j" - 2hat"k"`

Exercise 6.2 | Q 3 | Page 237

The volume of the parallelepiped whose coterminus edges are `7hat"i" + lambdahat"j" - 3hat"k", hat"i" + 2hat"j" - hat"k", -3hat"i" + 7hat"j" + 5hat"k"` is 90 cubic units. Find the value of λ 

Exercise 6.2 | Q 4 | Page 237

If `vec"a", vec"b", vec"c"` are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of `(vec"a" + vec"b") * (vec"b" xx vec"c") + (vec"b" + vec"c")* (vec"c" xx vec"a") + (vec"c" + vec"a") * (vec"a" xx vec"b")`

Exercise 6.2 | Q 5 | Page 237

Find the altitude of a parallelepiped determined by the vectors `vec"a" = - 2hat"i" + 5hat"j" + 3hat"k", vec"b" = hat"i" + 3hat"j" - 2hat"k"` and `vec"c" = - vec"i" + vec"j" + 4vec"k"` if the base is taken as the parallelogram determined by `vec"b"` and `vec"c"`

Exercise 6.2 | Q 6 | Page 237

Determine whether the three vectors `2hat"i" + 3hat"j" + hat"k", hat"i" - 2hat"j" + 2hat"k"` and `3hat"i" + hat"j" + 3hat"k"` are coplanar

Exercise 6.2 | Q 7 | Page 237

Ler `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = hat"i"` and `vec"c" = "c"_1hat"i" + "c"_2hat"j" + "c"_3hat"k"`. If c1 = 1 and c2 = 2. find c3 such that `vec"a", vec"b"` and `vec"c"` are coplanar

Exercise 6.2 | Q 8 | Page 238

If `vec"a" = hat"i" - hat"k", vec"b" = xhat"i" + hat"j" + (1 - x)hat"k", vec"c" = yhat"i" + xhat"j" + (1 + x - y)hat"k"`, show that  `[(vec"a", vec"b", vec"c")]` depends on neither x nor y

Exercise 6.2 | Q 9 | Page 238

If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k"` and `"c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, prove that c is the geometric mean of a and b

Exercise 6.2 | Q 10 | Page 238

Let `vec"a",  vec"b",  vec"c"` be three non-zero vectors such that `vec"c"` is a unit vector perpendicular to both `vec"a"` and `vec"b"`. If the angle between `vec"a"` and `vec"b"` is `pi/6`, show that `[(vec"a", vec"b", vec"c")]^2 = 1/4|vec"a"|^2|vec"b"|^2`

Exercise 6.3 [Page 242]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.3 [Page 242]

Exercise 6.3 | Q 1. (i) | Page 242

If `vec"a" = hat"i"  - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `(vec"a" xx vec"b") xx vec"c"`

Exercise 6.3 | Q 1. (ii) | Page 242

If `vec"a" = hat"i"  - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" xx (vec"b" xx vec"c")`

Exercise 6.3 | Q 2 | Page 242

For any vector `vec"a"`, prove that `hat"i" xx (vec"a" xx hat"i") + hat"j" xx (vec"a" xx hat"j") + hat"k" xx (vec"a" xx hat"k") = 2vec"a"`

Exercise 6.3 | Q 3 | Page 242

Prove that `[vec"a" - vec"b", vec"b" - vec"c", vec"c" - vec"a"]` = 0

Exercise 6.3 | Q 4. (i) | Page 242

If `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = 3hat"i" + 5hat"j" + 2hat"k", vec"c" = - hat"i" - 2hat"j" + 3hat"k"`, verify that `(vec"a" xx vec"b") xx vec"c" = (vec"a"*vec"c")vec"b" - (vec"b" * vec"c")vec"a"`

Exercise 6.3 | Q 4. (ii) | Page 242

If `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = 3hat"i" + 5hat"j" + 2hat"k", vec"c" = - hat"i" - 2hat"j" + 3hat"k"`, verify that `vec"a" xx (vec"b" xx vec"c") = (vec"a"*vec"c")vec"b" - (vec"a"*vec"b")vec"c"`

Exercise 6.3 | Q 5 | Page 242

`vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = -hat"i" + 2hat"j" - 4hat"k", vec"c" = hat"i" + hat"j" + hat"k"` then find the va;ue of `(vec"a" xx vec"b")*(vec"a" xx vec"c")`

Exercise 6.3 | Q 6 | Page 242

If `vec"a", vec"b", vec"c", vec"d"` are coplanar vectors, show that `(vec"a" xx vec"b") xx (vec"c" xx vec"d") = vec0`

Exercise 6.3 | Q 7 | Page 242

If `vec"a" = hat"i" + 2hat"j" + 3hat"k", vec"b" = 2hat"i" - hat"j" + hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"` and `vec"a" xx (vec"b" xx vec"c") = lvec"a" + "m"vec"b" + ""vec"c"`, find the values of l, m, n

Exercise 6.3 | Q 8 | Page 242

If `hat"a", hat"b", hat"c"` are three unit vectors such that `hat"b"` and `hat"c"` are non-parallel and `hat"a" xx (hat"b" xx hat"c") = 1/2 hat"b"`, find the angle between `hat"a"` and `hat"c"`

Exercise 6.4 [Page 249]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.4 [Page 249]

Exercise 6.4 | Q 1 | Page 249

Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector `4hat"i" + 3hat"j" - 7hat"k"` and parallel to the vector `2hat"i" - 6hat"j" + 7hat"k"`

Exercise 6.4 | Q 2 | Page 249

Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (– 2, 3, 4) and parallel to the straight line `(x - 1)/(-4) = (y + 3)/5 = (8 - z)/6`

Exercise 6.4 | Q 3 | Page 249

Find the points where the straight line passes through (6, 7, 4) and (8, 4, 9) cuts the xz and yz planes

Exercise 6.4 | Q 4 | Page 249

Find the direction cosines of the straight line passing through the points (5, 6, 7) and (7, 9, 13). Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points

Exercise 6.4 | Q 5. (i) | Page 249

Find the acute angle between the following lines.

`vec"r" = (4hat"i" - hat"j") + "t"(hat"i" + 2hat"j" - 2hat"k")`

Exercise 6.4 | Q 5. (ii) | Page 249

Find the acute angle between the following lines.

`(x + 4)/3 = (y - 7)/4 = (z + 5)/5, vec"r" = 4hat"k" + "t"(2hat"i" + hat"j" + hat"k")`

Exercise 6.4 | Q 5. (iii) | Page 249

Find the acute angle between the following lines.

2x = 3y = – z and 6x = – y = – 4z

Exercise 6.4 | Q 6 | Page 249

The vertices of ΔABC are A(7, 2, 1), 5(6, 0, 3), and C(4, 2, 4). Find ∠ABC

Exercise 6.4 | Q 7 | Page 249

f the straight line joining the points (2, 1, 4) and (a – 1, 4, – 1) is parallel to the line joining the points (0, 2, b – 1) and (5, 3, – 2) find the values of a and b

Exercise 6.4 | Q 8 | Page 249

If the straight lines `(x - 5)/(5"m" + 2) = (2 - y)/5 = (1 - z)/(-1)` and x = `(2y + 1)/(4"m") = (1 - z)/(-3)` are perpendicular to ech other find the  value of m

Exercise 6.4 | Q 9 | Page 249

Show that the points (2, 3, 4), (– 1, 4, 5) and (8, 1, 2) are collinear

Exercise 6.5 [Page 255]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.5 [Page 255]

Exercise 6.5 | Q 1 | Page 255

Find the parametric form of vector equation and Cartesian equations of straight line passing through (5, 2, 8) and is perpendicular to the straight lines `vec"r" = (hat"i" + hat"j" - hat"k") + "s"(2hat"i" - 2hat"j" + hat"k")` and `vec"r" = (2hat"i" - hat"j" - 3hat"k") + "t"(hat"i" + 2hat"j" + 2hat"k")`

Exercise 6.5 | Q 2 | Page 255

Show that the lines `vec"r" = (6hat"i" + hat"j" + 2hat"k") + "s"(hat"i" + 2hat"j" - 3hat"k")` and `vec"r" = (3hat"i" + 2hat"j" - 2hat"k") + "t"(2hat"i" + 4hat"j" - 5hat"k")` are skew lines and hence find the shortest distance between them

Exercise 6.5 | Q 3 | Page 255

If the two lines `(x - 1)/2 = (y + 1)/3 = (z - 1)/4` and `(x - 3)/1 = (y - "m")/2` = z intersect at a point, find the value of m

Exercise 6.5 | Q 4 | Page 255

Show that the lines `(x - 3)/3 = (y - 3)/(-1), z - 1` = 0 and `(x - 6)/2 = (z - 1)/3, y - 2` = 0 intersect. Aslo find the point of intersection

Exercise 6.5 | Q 5 | Page 255

Show that the straight lines x + 1 = 2y = – 12z and x = y + 2 = 6z – 6 are skew and hence find the shortest distance between them

Exercise 6.5 | Q 6 | Page 255

Find the parametric form of vector equation of the straight line passing through (−1, 2, 1) and parallel to the straight line `vec"r" = (2hat"i" + 3hat"j" - hat"k") + "t"(hat"i" - 2hat"j" + hat"k")` and hence find the shortest distance between the lines

Exercise 6.5 | Q 7 | Page 255

Find the foot of the perpendicular drawn: from the point (5, 4, 2) to the line `(x + 1)/2 = (y - 3)/3 = (z - 1)/(-1)`. Also, find the equation of the perpendicular

Exercise 6.6 [Page 259]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.6 [Page 259]

Exercise 6.6 | Q 1 | Page 259

Find a parametric form of vector equation of a plane which is at a distance of 7 units from t the origin having 3, – 4, 5 as direction ratios of a normal to it

Exercise 6.6 | Q 2 | Page 259

Find the direction cosines of the normal to the plane 12x + 3y – 4z = 65. Also find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin

Exercise 6.6 | Q 3 | Page 259

Find the vector and Cartesian equation of the plane passing through the point with position vector `2hat"i" + 6hat"j" + 3hat"k"` and normal to the vector `hat"i" + 3hat"j" + 5hat"k"`

Exercise 6.6 | Q 4 | Page 259

A plane passes through the point (− 1, 1, 2) and the normal to the plane of magnitude `3sqrt(3)` makes equal acute angles with the coordinate axes. Find the equation of the plane

Exercise 6.6 | Q 5 | Page 259

Find the intercepts cut off by the plane `vec"r"*(6hat"i" + 45hat"j" - 3hat"k")` = 12 on the coordinate axes

Exercise 6.6 | Q 6 | Page 259

If a plane meets the co-ordinate axes at A, B, C such that the centroid of the triangle ABC is the point (u, v, w), find the equation of the plane

Exercise 6.7 [Page 263]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.7 [Page 263]

Exercise 6.7 | Q 1 | Page 263

Find the non-parametric form of vector equation and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to thestraight lines `(x - 1)/2 = (y + 1)/3 = (x - 3)/1` and `(x + 3)/2 = (y - 3)/(-5) = (z + 1)/(-3)`

Exercise 6.7 | Q 2 | Page 263

Find the parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9

Exercise 6.7 | Q 3 | Page 263

Find the parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, – 2, 3) and parallel to the straight line passing through the points (2, 1, – 3) and (– 1, 5, – 8)

Exercise 6.7 | Q 4 | Page 263

Find the non-parametric form of vector equation and cartesian equation of the plane passing through the point (1, − 2, 4) and perpendicular to the plane x + 2y − 3z = 11 and parallel to the line `(x + 7)/3 = (y + 3)/(-1) = z/1`

Exercise 6.7 | Q 5 | Page 263

Find the parametric form of vector equation, and Cartesian equations of the plane containing the line `vec"r" = (hat"i" - hat"j" + 3hat"k") + "t"(2hat"i" - hat"j" + 4hat"k")` and perpendicular to plane `vec"r"*(hat"i" + 2hat"j" + hat"k")` = 8

Exercise 6.7 | Q 6 | Page 263

Find the parametric vector, non-parametric vector and Cartesian form of the equation of the plane passing through the point (3, 6, – 2), (– 1, – 2, 6) and (6, 4, – 2)

Exercise 6.7 | Q 7 | Page 263

Find the non-parametric form of vector equation and Cartesian equations of the plane `vec"r" = (6hat"i" - hat"j" + hat"k") + "s"(-hat"i" + 2hat"j" + hat"k") + "t"(-5hat"i" - 4hat"j" - 5hat"k")`

Exercise 6.8 [Page 266]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.8 [Page 266]

Exercise 6.8 | Q 1 | Page 266

Show that the straight lines `vec"r" = (5hat"i" + 7hat"j" - 3hat"k") + "s"(4hat"i" + 4hat"j" - 5hat"k")` and `vec"r"(8hat"i" + 4hat"j" + 5hat"k") + "t"(7hat"i" + hat"j" + 3hat"k")` are coplanar. Find the vector equation of the plane in which they lie

Exercise 6.8 | Q 2 | Page 266

Show that the lines `(x - 2)/1 = (y - 3)/1 = (z - 4)/3` and `(x - 1)/(-3) = (y - 4)/2 = (z - 5)/1` are coplanar. Also, find the plane containing these lines

Exercise 6.8 | Q 3 | Page 266

If the straight lines `(x - 1)/1 - (y - 2)/2 = (z - 3)/"m"^2` and `(x - 3)/5 = (y - 2)/"m"^2 = (z - 1)/2` are coplanar, find the distinct real values of m

Exercise 6.8 | Q 4 | Page 266

If the straight lines `(x - 1)/2 = (y + 1)/lambda = z/2` and `(x + 1)/5 = (y + 1)/2 = z/lambda` are coplanar, find λ and equations of the planes containing these two lines

Exercise 6.9 [Page 276]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.9 [Page 276]

Exercise 6.9 | Q 1 | Page 276

Find the equation of the plane passing through the line of intersection of the planes `vec"r"*(2hat"i" - 7hat"j" + 4hat"k")` = 3 and 3x – 5y + 4z + 11 = 0, and the point (– 2, 1, 3)

Exercise 6.9 | Q 2 | Page 276

Find the equation of the plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x – y + z = 3 and at a distance `2/sqrt(3)` from the point (3, 1, –1)

Exercise 6.9 | Q 3 | Page 276

Find the angle between the line `vec"r" = (2hat"i" - hat"j" + hat"k") + "t"(hat"i" + 2hat"j" - 2hat"k")` and the plane `vec"r"*(6hat"i" + 3hat"j" + 2hat"k")` = 8

Exercise 6.9 | Q 4 | Page 276

Find the angle between the planes `vec"r"*(hat"i" + hat"j" - 2hat"k")` = 3 and 2x – 2y + z = 2

Exercise 6.9 | Q 5 | Page 276

Find the equation of the plane which passes through the point (3, 4, –1) and is parallel to the plane 2x – 3y + 5z + 7 = 0. Also, find the distance between the two planes

Exercise 6.9 | Q 6 | Page 276

Find the length of the perpendicular from the point (1, – 2, 3) to the plane x – y + z = 5

Exercise 6.9 | Q 7 | Page 276

Find the point of intersection of the line with the plane (x – 1) = `y/2` = z + 1 with the plane 2x – y – 2z = 2. Also, the angle between the line and the plane

Exercise 6.9 | Q 8 | Page 276

Find the co-ordinates of the foot of the perpendicular and length of the perpendicular from the point (4, 3, 2) to the plane x + 2y + 3z = 2.

Exercise 6.10 [Pages 276 - 278]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 6 Applications of Vector AlgebraExercise 6.10 [Pages 276 - 278]

MCQ

Exercise 6.10 | Q 1 | Page 276

Choose the correct alternative:

If `vec"a"` and `vec"b"` are parallel vectors, then `[vec"a", vec"c", vec"b"]` is equal to

  • 2

  • – 1

  • 1

  • 0

Exercise 6.10 | Q 2 | Page 276

Choose the correct alternative:

If a vector `vecalpha` lies in the plane of `vecbeta` and `vecϒ`, then

  • `[vecalpha, vecbeta, vecϒ]` = 1

  • `[vecalpha, vecbeta, vecϒ]` = – 1

  • `[vecalpha, vecbeta, vecϒ]` = 0

  • `[vecalpha, vecbeta, vecϒ]` = 2

Exercise 6.10 | Q 3 | Page 276

Choose the correct alternative:

If `vec"a"*vec"b" = vec"b"*vec"c" = vec"c"*vec"a"` = 0, then the value of `[vec"a", vec"b", vec"c"]` is

  • `|vec"a"||vec"b"||vec"c"|`

  • `1/3|vec"a"||vec"b"||vec"c"|`

  • 1

  • – 1

Exercise 6.10 | Q 4 | Page 276

Choose the correct alternative:

If `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" xx (vec"b" xx vec"c")` is equal to

  • `vec"a"`

  • `vec"b"`

  • `vec"c"`

  • `vec0`

Exercise 6.10 | Q 5 | Page 276

Choose the correct alternative:

If `[vec"a", vec"b", vec"c"]` = 1, then the value of `(vec"a"*(vec"b" xx vec"c"))/((vec"c" xx vec"a")*vec"b") + (vec"b"*(vec"c" xx vec"a"))/((vec"a" xx vec"b")*vec"c") + (vec"c"*(vec"a" xx vec"b"))/((vec"c" xx vec"b")*vec"a")` is

  • 1

  • – 1

  • 2

  • 3

Exercise 6.10 | Q 6 | Page 276

Choose the correct alternative:

The volume of the parallelepiped with its edges represented by the vectors `hat"i" + hat"j", hat"i" + 2hat"j", hat"i" + hat"j" + pihat"k"` is

  • `pi/2`

  • `pi/3`

  • π

  • `pi/4`

Exercise 6.10 | Q 7 | Page 277

Choose the correct alternative:

If `vec"a"` and `vec"b"` are unit vectors such that `[vec"a", vec"b", vec"a" xx vec"b"] = 1/4`, are unit vectors such that `vec"a"` nad `vec"b"` is

  • `pi/6`

  • `pi/4`

  • `pi/3`

  • `pi/2`

Exercise 6.10 | Q 8 | Page 277

Choose the correct alternative:

If `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = hat"i" + hat"j", vec"c" = hat"i"` and `(vec"a" xx vec"b")vec"c" - lambdavec"a" + muvec"b"` then the value of λ + µ is

  • 0

  • 1

  • 6

  • 3

Exercise 6.10 | Q 9 | Page 277

Choose the correct alternative:

If `vec"a", vec"b", vec"c"` are non-coplanar, non-zero vectors `[vec"a", vec"b", vec"c"]` = 3, then `{[[vec"a" xx vec"b", vec"b" xx vec"c", vec"c" xx vec"a"]]}^2` is equal to

  • 81

  • 9

  • 27

  • 18

Exercise 6.10 | Q 10 | Page 277

Choose the correct alternative:

If `vec"a", vec"b", vec"c"` are three non-coplanar vectors such that `vec"a" xx (vec"b" xx vec"c") = (vec"b" + vec"c")/sqrt(2)` then the angle between `vec"a"` and `vec"b"` is

  • `pi/2`

  • `(3pi)/4`

  • `pi/4`

  • π

Exercise 6.10 | Q 11 | Page 277

Choose the correct alternative:

If the volume of the parallelepiped with `vec"a" xx vec"b", vec"b" xx vec"c", vec"c" xx vec"a"` as coterminous edges is 8 cubic units, then the volume of the parallelepiped with `(vec"a" xx vec"b") xx (vec"b" xx vec"c"), (vec"b" xx vec"c") xx (vec"c" xx vec"a")` and `(vec"c" xx vec"a") xx (vec"a" xx vec"b")` as coterminous edges is

  • 64 cubic units

  • 512 cubic units

  • 64 cubic units

  • 24 cubic units

Exercise 6.10 | Q 12 | Page 277

Choose the correct alternative:

Consider the vectors  `vec"a", vec"b", vec"c", vec"d"` such that `(vec"a" xx vec"b") xx (vec"c" xx vec"d") = vec0`. Let P1 and P2 be the planes determined by the pairs of vectors `vec"a", vec"b"` and `vec'c", vec"d"` respectively. Then the angle between P1 and P2 is

  • 45°

  • 60°

  • 90°

Exercise 6.10 | Q 13 | Page 277

Choose the correct alternative:

I`vec"a" xx  (vec"b" xx vec"c") = (vec"a" xx vec"b") xx vec"c"`, where `vec"a", vec"b", vec"c"` are any three vectors such that `vec"b"*vec"c" ≠ 0` and `vec"a"*vec"b" ≠ 0`, then `vec"a"` and `vec"c"` are

  • perpendicular

  • parallel

  • inclined at an angle `pi/3`

  • inclined at an angle `pi/6`

Exercise 6.10 | Q 14 | Page 277

Choose the correct alternative:

If `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = hat"i" + 2hat"j" - 5hat"k", vec"c" = 3hat"i" + 5hat"j" - hat"k"`, then a vector perpendicular to `vec"a"` and lies in the plane containing `vec"b"` and `vec"c"` is 

  • `-17hat"i" + 21hat"j" - 97hat"k"`

  • `7hat"i"+  21hat"j"  123hat"k"`

  • `-17hat"i" - 21hat"j" + 97hat"k"`

  • `-17hat"i" - 21hat"j" - 97hat"k"`

Exercise 6.10 | Q 15 | Page 277

Choose the correct alternative:

The angle between the lines `(x - 2)/3 = (y + 1)/(-2)`, z = 2 ad `(x - 1)/1 = (2y + 3)/3 = (z + 5)/2` is

  • `pi/6`

  • `pi/4`

  • `pi/3`

  • `pi/2`

Exercise 6.10 | Q 16 | Page 278

Choose the correct alternative:

If the line `(x  - )/3 = (y - 1)/(-5) = (x + 2)/2` lies in the plane x + 3y – αz + ß = 0 then (α + ß) is

  • (– 5, 5)

  • (– 6, 7)

  • (5, – 5)

  • (6, – 7)

Exercise 6.10 | Q 17 | Page 278

Choose the correct alternative:

The angle between the line `vec"r" = (hat"i" + 2hat"j" - 3hat"k") + "t"(2hat"i" + hat"j" - 2hat"k")` and the plane `vec"r"(hat"i" + hat"j") + 4` = 0 is

  • 30°

  • 45°

  • 90°

Exercise 6.10 | Q 18 | Page 278

Choose the correct alternative:

The coordinates of the point where the line `vec"r" = (6hat"i" - hat"j" - 3hat"k") + "t"(- hat"i" + 4hat"k")` meets the plane `vec"r"*(hat"i" + hat"j" - hat"k")` = 3 are

  • (2, 1, 0)

  • (7, – 1, – 7)

  • (1, 2, – 6)

  • (5, –1, 1)

Exercise 6.10 | Q 19 | Page 278

Choose the correct alternative:

Distance from the origin to the plane 3x – 6y + 2z + 7 = 0 is

  • 0

  • 1

  • 2

  • 3

Exercise 6.10 | Q 20 | Page 278

Choose the correct alternative:

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is

  • `sqrt(7)/(2sqrt(2))`

  • `7/2`

  • `sqrt(7)/2`

  • `7/(2sqrt(2))`

Exercise 6.10 | Q 21 | Page 278

Choose the correct alternative:

If the direction cosines of a line are `1/"c", 1/"c", 1/"c"`

  • c = ± 3

  • c = `+-  sqrt(3)`

  • c > 0

  • 0 < c < 1

Exercise 6.10 | Q 22 | Page 278

Choose the correct alternative:

The vector equation `vec"r" = (hat"i" - hat"j" - hat"k") + "t"(6hat"i" - hat"k")` represents a straight line passing through the points

  • (0, 6, −1) and (1, − 2, −1)

  • (0, 6, −1) − and (−1,− 4, −2)

  • (1, −2, −1) and (1, 4, −2)

  • (1, −2, −1) and (0, −6, 1)

Exercise 6.10 | Q 23 | Page 278

Choose the correct alternative:

If 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

  • ± 3

  • ± 6

  • – 3, 9

  • 3, – 9

Exercise 6.10 | Q 24 | Page 278

Choose the correct alternative:

If the planes `vec"r"(2hat"i" - lambdahat"j" + hatk")` =  and `vec"r"(4hat"i" + hat"j" - muhat"k")` = 5 are parallel, then the value of λ and µ are

  • `1/2, 2`

  • `- 1/2, 2`

  • `- 1/2, -2`

  • `1/2, 2`

Exercise 6.10 | Q 25 | Page 278

Choose the correct alternative:

If the length of the perpendicular from the origin to the plane 2x + 3y + λz = 1, λ > 0 is `1/5, then the value of λ is

  • `2sqrt(3)`

  • `3sqrt(2)`

  • 0

  • 1

Chapter 6: Applications of Vector Algebra

Exercise 6.1Exercise 6.2Exercise 6.3Exercise 6.4Exercise 6.5Exercise 6.6Exercise 6.7Exercise 6.8Exercise 6.9Exercise 6.10
Class 12th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 6 - Applications of Vector Algebra

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 6 (Applications of Vector Algebra) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 12th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 6 Applications of Vector Algebra are Introduction to Applications of Vector Algebra, Geometric Introduction to Vectors, Scalar Product and Vector Product, Scalar Triple Product of Vectors, Vector Triple Product, Jacobi’S Identity and Lagrange’S Identity, Application of Vectors to 3-dimensional Geometry, Different Forms of Equation of a Plane, Image of a Point in a Plane, Meeting Point of a Line and a Plane.

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