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

## Chapter 10: Vector Algebra

Solved ExamplesExercise
Solved Examples [Pages 206 - 214]

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

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"

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 2vec"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]

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.

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

## 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.

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