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NCERT solutions for Mathematics Exemplar Class 12 chapter 10 - Vector Algebra [Latest edition]

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Chapter 10: Vector Algebra

Solved ExamplesExercise
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Solved Examples [Pages 206 - 214]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 10 Vector AlgebraSolved Examples [Pages 206 - 214]

Short Answer

Solved Examples | Q 1 | Page 206

Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + 2hat"k"` and `vec"b" = -hat"i" + hat"j" + 3hat"k"`.

Solved Examples | Q 2 | Page 207

Find a vector of magnitude 11 in the direction opposite to that of `vec"PQ"` where P and Q are the points (1, 3, 2) and (–1, 0, 8), respectively.

Solved Examples | Q 3.(i) | Page 207

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors `vec"OP" = 2vec"a" + vec"b"` and `vec"OQ" = vec"a" - 2vec"b"`, respectively, in the ratio 1:2 internally

Solved Examples | Q 3.(ii) | Page 207

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors `vec"OP" = 2vec"a" + vec"b"` and `vec"OQ" = vec"a" - 2vec"b"`, respectively, in the ratio 1:2 externally

Solved Examples | Q 4 | Page 208

If the points (–1, –1, 2), (2, m, 5) and (3,11, 6) are collinear, find the value of m.

Solved Examples | Q 5 | Page 208

Find a vector `vec"r"` of magnitude `3sqrt(2)` units which makes an angle of `pi/4` and `pi/2` with y and z-axes, respectively.

Solved Examples | Q 6 | Page 209

If `vec"a" = 2hat"i" - hat"j" + hat"k", vec"b" = hat"i" + hat"j" - 2hat"k"` and `vec"c" = hat"i" + 3hat"j" - hat"k"`, find `lambda` such that `vec"a"` is perpendicular to `lambdavec"b" + vec"c"`.

Solved Examples | Q 7 | Page 209

Find all vectors of magnitude `10sqrt(3)` that are perpendicular to the plane of `hat"i" + 2hat"j" + hat"k"` and `-hat"i" + 3hat"j" + 4hat"k"`

Long Answer

Solved Examples | Q 8 | Page 210

Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.

Solved Examples | Q 9 | Page 211

Prove that in a ∆ABC,  `sin"A"/"a" = sin"B"/"b" = sin"C"/"c"`, where a, b, c represent the magnitudes of the sides opposite to vertices A, B, C, respectively.

Objective Type Questions from 10 to 21

Solved Examples | Q 10 | Page 211

The magnitude of the vector `6hat"i" + 2hat"j" + 3hat"k"` is ______.

  • 5

  • 7

  • 12

  • 1

Solved Examples | Q 11 | Page 212

The position vector of the point which divides the join of points with position vectors `vec"a" + vec"b"` and 2`vec"a" - vec"b"` in the ratio 1:2 is ______.

  • `(3vec"a" + 2vec"b")/3`

  • `vec"a"`

  • `(5vec"a" - vec"b")/3`

  • `(4vec"a" + vec"b")/3`

Solved Examples | Q 12 | Page 212

The vector with initial point P (2, –3, 5) and terminal point Q(3, –4, 7) is ______.

  • `hat"i" - hat"j" + 2hat"k"`

  • `5hat"i" - 7hat"j" + 12hat"k"`

  • `-hat"i" + hat"j" - 2hat"k"`

  • None of these

Solved Examples | Q 13 | Page 212

The angle between the vectors `hat"i" - hat"j"` and `hat"j" - hat"k"` is ______.

  • `pi/3`

  • `(2pi)/3`

  • `(-pi)/3`

  • `(5pi)/6`

Solved Examples | Q 14 | Page 212

The value of λ for which the two vectors `2hat"i" - hat"j" + 2hat"k"` and `3hat"i" + lambdahat"j" + hat"k"` are perpendicular is ______.

  • 2

  • 4

  • 6

  • 8

Solved Examples | Q 15 | Page 213

The area of the parallelogram whose adjacent sides are `hat"i" + hat"k"` and `2hat"i" + hat"j" + hat"k"` is ______.

  • `sqrt(2)`

  • `sqrt(3)`

  • 3

  • 4

Solved Examples | Q 16 | Page 213

If `|vec"a"|` = 8, `|vec"b"|` = 3 and `|vec"a" xx vec"b"|` = 12, then value of `vec"a" * vec"b"` is ______.

  • `6sqrt(3)`

  • `8sqrt(3)`

  • `12sqrt(3)`

  • None of these

Solved Examples | Q 17 | Page 213

The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC, respectively of a ∆ABC. The length of the median through A is ______.

  • `sqrt(34)/2`

  • `sqrt(48)/2`

  • `sqrt(18)`

  • None of these

Solved Examples | Q 18 | Page 213

The projection of vector `vec"a" = 2hat"i" - hat"j" + hat"k"` along `vec"b" = hat"i" + 2hat"j" + 2hat"k"` is ______.

  • `2/3`

  • `1/3`

  • 2

  • `sqrt(6)`

Solved Examples | Q 19 | Page 214

If `vec"a"` and `vec"b"` are unit vectors, then what is the angle between `vec"a"` and `vec"b"` for `sqrt(3)  vec"a" - vec"b"` to be a unit vector?

  • 30°

  • 45°

  • 60°

  • 90°

Solved Examples | Q 20 | Page 214

The unit vector perpendicular to the vectors `hat"i" - hat"j"` and `hat"i" + hat"j"` forming a right handed system is ______.

  • `hat"k"`

  • `-hat"k"`

  • `(hat"i" - hat"j")/sqrt(2)`

  • `(hat"i" + hat"j")/sqrt(2)`

Solved Examples | Q 21 | Page 214

If `|vec"a"|` = 3 and –1 ≤ k ≤ 2, then `|"k"vec"a"|` lies in the interval ______.

  • [0, 6]

  • [– 3, 6]

  • [3, 6]

  • [1, 2]

Exercise [Pages 215 - 219]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 10 Vector AlgebraExercise [Pages 215 - 219]

Short Answer

Exercise | Q 1 | Page 215

Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + hat"k"` and `vec"b" = 2hat"j" + hat"k"`.

Exercise | Q 2.(i) | Page 215

If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `6vec"b"`

Exercise | Q 2.(ii) | Page 215

If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `hat"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `2vec"a" - vec"b"`

Exercise | Q 3 | Page 215

Find a unit vector in the direction of `vec"PQ"`, where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively

Exercise | Q 4 | Page 215

If `vec"a"` and `vec"b"` are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.

Exercise | Q 5 | Page 215

Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.

Exercise | Q 6 | Page 215

A vector `vec"r"` is inclined at equal angles to the three axes. If the magnitude of `vec"r"` is `2sqrt(3)` units, find `vec"r"`.

Exercise | Q 7 | Page 215

A vector `vec"r"` has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of `vec"r"`, given that `vec"r"` makes an acute angle with x-axis.

Exercise | Q 8 | Page 215

Find a vector of magnitude 6, which is perpendicular to both the vectors `2hat"i" - hat"j" + 2hat"k"` and `4hat"i" - hat"j" + 3hat"k"`.

Exercise | Q 9 | Page 215

Find the angle between the vectors `2hat"i" - hat"j" + hat"k"` and `3hat"i" + 4hat"j" - hat"k"`.

Exercise | Q 10 | Page 215

If `vec"a" + vec"b" + vec"c"` = 0, show that `vec"a" xx vec"b" = vec"b" xx vec"c" = vec"c" xx vec"a"`. Interpret the result geometrically?

Exercise | Q 11 | Page 215

Find the sine of the angle between the vectors `vec"a" = 3hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" - 2hat"j" + 4hat"k"`.

Exercise | Q 12 | Page 216

If A, B, C, D are the points with position vectors `hat"i" + hat"j" - hat"k", 2hat"i" - hat"j" + 3hat"k", 2hat"i" - 3hat"k", 3hat"i" - 2hat"j" + hat"k"`, respectively, find the projection of `vec"AB"` along `vec"CD"`.

Exercise | Q 13 | Page 216

Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).

Exercise | Q 14 | Page 216

Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.

Long Answer

Exercise | Q 15 | Page 216

Prove that in any triangle ABC, cos A = `("b"^2 + "c"^2 - "a"^2)/(2"bc")`, where a, b, c are the magnitudes of the sides opposite to the vertices A, B, C, respectively.

Exercise | Q 16 | Page 216

If `vec"a", vec"b", vec"c"` determine the vertices of a triangle, show that `1/2[vec"b" xx vec"c" + vec"c" xx vec"a" + vec"a" xx vec"b"]` gives the vector area of the triangle. Hence deduce the condition that the three points `vec"a", vec"b", vec"c"` are collinear. Also find the unit vector normal to the plane of the triangle.

Exercise | Q 17 | Page 216

Show that area of the parallelogram whose diagonals are given by `vec"a"` and `vec"b"` is `(|vec"a" xx vec"b"|)/2`. Also find the area of the parallelogram whose diagonals are `2hat"i" - hat"j" + hat"k"` and `hat"i" + 3hat"j" - hat"k"`.

Exercise | Q 18 | Page 216

If `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"j" - hat"k"`, find a vector `vec"c"` such that `vec"a" xx vec"c" = vec"b"` and `vec"a"*vec"c"` = 3.

Objective Type Questions from 19 to 33

Exercise | Q 19 | Page 216

The vector in the direction of the vector `hat"i" - 2hat"j" + 2hat"k"` that has magnitude 9 is ______.

  • `hat"i" - 2hat"j" + 2hat"k"`

  • `(hat"i" - 2hat"j" + 2hat"k")/3`

  • `3(hat"i" - 2hat"j" + 2hat"k")`

  • `9(hat"i" - 2hat"j" + 2hat"k")`

Exercise | Q 20 | Page 217

The position vector of the point which divides the join of points `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1 is ______.

  • `(3vec"a" - 2vec"b")/2`

  • `(7vec"a" - 8vec"b")/4`

  • `(3vec"a")/4`

  • `(5vec"a")/4`

Exercise | Q 21 | Page 217

The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is ______.

  • `-hat"i" + 12hat"j" + 4hat"k"`

  • `5hat"i" + 2hat"j" - 4hat"k"`

  • `-5hat"i" + 2hat"j" + 4hat"k"`

  • `hat"i" + hat"j" + hat"k"`

Exercise | Q 22 | Page 217

The angle between two vectors `vec"a"` and `vec"b"` with magnitudes `sqrt(3)` and 4, respectively, and `vec"a" * vec"b" = 2sqrt(3)` is ______.

  • `pi/6`

  • `pi/3`

  • `pi/2`

  • `(5pi)/2`

Exercise | Q 23 | Page 217

Find the value of λ such that the vectors `vec"a" = 2hat"i" + lambdahat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"` are orthogonal ______.

  • 0

  • 1

  • `3/2`

  • `- 5/2`

Exercise | Q 24 | Page 217

The value of λ for which the vectors `3hat"i" - 6hat"j" + hat"k"` and `2hat"i" - 4hat"j" + lambdahat"k"` are parallel is ______.

  • `2/3`

  • `3/2`

  • `5/2`

  • `2/5`

Exercise | Q 25 | Page 217

The vectors from origin to the points A and B are `vec"a" = 2hat"i" - 3hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + 3hat"j" + hat"k"`, respectively, then the area of triangle OAB is ______.

  • 340

  • `sqrt(25)`

  • `sqrt(229)`

  • `1/2sqrt(229)`

Exercise | Q 26 | Page 218

For any vector `vec"a"`, the value of `(vec"a" xx hat"i")^2 + (vec"a" xx hat"j")^2 + (vec"a" xx hat"k")^2` is equal to ______.

  • `vec"a"^2`

  • `3vec"a"^2`

  • `4vec"a"^2`

  • `2vec"a"^2`

Exercise | Q 27 | Page 218

If `|vec"a"|` = 10, `|vec"b"|` = 2 and `vec"a".vec"b"` = 12, then value of `|vec"a" xx vec"b"|` is ______.

  • 5

  • 10

  • 14

  • 16

Exercise | Q 28 | Page 218

The vectors `lambdahat"i" + hat"j" + 2hat"k", hat"i" + lambdahat"j" - hat"k"` and `2hat"i" - hat"j" + lambdahat"k"` are coplanar if ______.

  • λ = –2

  • λ = 0

  • λ = 1

  • λ = – 1

Exercise | Q 29 | Page 218

If `vec"a", vec"b", vec"c"` are unit vectors such that `vec"a" + vec"b" + vec"c"` = 0, then the value of `vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a"` is ______.

  • 1

  • 3

  • ` -3/2`

  • None of these

Exercise | Q 30 | Page 218

Projection vector of `vec"a"` on `vec"b"` is ______.

  • `((vec"a"*vec"b")/|vec"b"|^2)vec"b"`

  • `(vec"a"*vec"b")/|vec"b"|`

  • `(vec"a"*vec"b")/|vec"a"|`

  • `((vec"a"*vec"b")/|vec"a"|^2)vec"b"`

Exercise | Q 31 | Page 218

If `vec"a", vec"b", vec"c"` are three vectors such that `vec"a" + vec"b" + vec"a" = vec0` and `|vec"a"|` = 2, `|vec"b"|` = 3, `|vec"c"|` = 5, then value of `vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a"` is ______.

  • 0

  • 1

  • – 19

  • 38

Exercise | Q 32 | Page 218

If `|vec"a"|` = 4 and −3 ≤ λ ≤ 2, then the range of `|lambdavec"a"|` is ______.

  • [0, 8]

  • [–12, 8]

  • [0, 12]

  • [8, 12]

Exercise | Q 33 | Page 218

The number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.

  • One

  • Two

  • Three

  • Infinite

Fill in the blanks 34 to 40

Exercise | Q 34 | Page 218

The vector `vec"a" + vec"b"` bisects the angle between the non-collinear vectors `vec"a"` and `vec"b"` if ______.

Exercise | Q 35 | Page 219

If `vec"r" * vec"a" = 0, vec"r" * vec"b" = 0` and `vec"r" * vec"c" = 0` for some non-zero vector `vec"r"`, then the value of `vec"a" * (vec"b" xx vec"c")` is ______.

Exercise | Q 36 | Page 219

The vectors `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"` and `vec"b" = -hat"i" - 2hat"k"` are the adjacent sides of a parallelogram. The acute angle between its diagonals is ______.

Exercise | Q 37 | Page 219

The values of k for which `|"k"vec"a"| < |vec"a"|` and `"k"vec"a" + 1/2 vec"a"` is parallel to `vec"a"` holds true are ______.

Exercise | Q 38 | Page 219

The value of the expression `|vec"a" xx vec"b"|^2 + (vec"a".vec"b")^2` is ______.

Exercise | Q 39 | Page 219

If `|vec"a" xx vec"b"|^2 + |vec"a".vec"b"|^2` = 144 and `|vec"a"|` = 4, then `|vec"b"|` is equal to ______.

Exercise | Q 40 | Page 219

If `vec"a"` is any non-zero vector, then `(vec"a" .hat"i")hat"i" + (vec"a".hat"j")hat"j" + (vec"a".hat"k")hat"k"` equals ______.

State True or False in the following

Exercise | Q 41 | Page 219

If `|vec"a"| = |vec"b"|`, then necessarily it implies `vec"a" = +- vec"b"`.

  • True

  • False

Exercise | Q 42 | Page 219

Position vector of a point P is a vector whose initial point is origin.

  • True

  • False

Exercise | Q 43 | Page 219

If `|vec"a" + vec"b"| = |vec"a" - vec"b"|`, then the vectors `vec"a"` and `vec"b"` are orthogonal.

  • True

  • False

Exercise | Q 44 | Page 219

The formula `(vec"a" + vec"b")^2 = vec"a"^2 + vec"b"^2 + 2vec"a" xx vec"b"` is valid for non-zero vectors `vec"a"` and `vec"b"`

  • True

  • False

Exercise | Q 45 | Page 219

If `vec"a"` and `vec"b"` are adjacent sides of a rhombus, then `vec"a" * vec"b"` = 0

  • True

  • False

Chapter 10: Vector Algebra

Solved ExamplesExercise
Mathematics Exemplar Class 12 - Shaalaa.com

NCERT solutions for Mathematics Exemplar Class 12 chapter 10 - Vector Algebra

NCERT solutions for Mathematics Exemplar Class 12 chapter 10 (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 CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 12 chapter 10 Vector Algebra are Direction Cosines, Properties of Vector Addition, Geometrical Interpretation of Scalar, Scalar Triple Product of Vectors, Vector (Or Cross) Product of Two Vectors, Scalar (Or Dot) Product of Two Vectors, Position Vector of a Point Dividing a Line Segment in a Given Ratio, Multiplication of a Vector by a Scalar, Addition of Vectors, Vectors and Their Types, Introduction of Vector, Magnitude and Direction of a Vector, Basic Concepts of Vector Algebra, Components of a Vector, Section Formula, Vector Joining Two Points, Vectors Examples and Solutions, Projection of a Vector on a Line, Introduction of Product of Two Vectors.

Using NCERT Class 12 solutions Vector Algebra exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

Get the free view of chapter 10 Vector Algebra Class 12 extra questions for Mathematics Exemplar Class 12 and can use Shaalaa.com to keep it handy for your exam preparation

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