NCERT Solutions for Class 12 Maths Chapter 10 – Free PDF Download

The NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra are given here, where the students learn about the difference between a scalar and a vector quantity, their properties, operations of vectors, etc. The topic has an important role in helping students score high marks not only in board exams but also in competitive exams. The NCERT Solutions for Class 12 Maths are important to prepare for the various problems asked during the Class 12 Maths board examination.

These Class 12 Maths NCERT Solutions for Vector Algebra are very easy-to-understand and are in accordance with the latest CBSE Syllabus 2023-24. Solving through different exercises and problem sets gives students the confidence to perform better in the CBSE exams. Students can also avail of NCERT Solutions for Class 12 Maths Chapter 10 in a downloadable PDF for free to practise them offline as well.

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra

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NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

Exercise 10.1 Page No: 428

1. Represent graphically a displacement of 40 km, 30° east of north.

Solution:

The vector
represents the displacement of 40 km, 30^o east of north.

2. Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north-west (iii) 40°

(iv) 40 watt (v) 10^–19 coulomb (vi) 20 m/s²

Solution:

(i) 10 kg is a scalar quantity because it has only magnitude.

(ii) 2 meters north-west is a vector quantity as it has both magnitude and direction.

(iii) 40° is a scalar quantity as it has only magnitude.

(iv) 40 watt is a scalar quantity as it has only magnitude.

(v) 10^–19 coulomb is a scalar quantity as it has only magnitude.

(vi) 20 m/s² is a vector quantity as it has both magnitude and direction.

3. Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

Solution:

(i) Time period is a scalar quantity as it has only magnitude.

(ii) Distance is a scalar quantity as it has only magnitude.

(iii) Force is a vector quantity as it has both magnitude and direction.

(iv) Velocity is a vector quantity as it has both magnitude as well as direction.

(v) Work done is a scalar quantity as it has only magnitude.

4. In Figure, identify the following vectors.

(i) Coinitial (ii) Equal (iii) Collinear but not equal

Solution:

directions are not the same.

5. Answer the following as true or false.

(i)  andare collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having the same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

Solution:

(i) True.

Vectors
 and
are parallel to the same line.

(ii) False.

Collinear vectors are those vectors that are parallel to the same line.

(iii) False.

Two vectors having the same magnitude need not necessarily be parallel to the same line.

(iv) False.

Only if the magnitude and direction of two vectors are the same, regardless of the positions of their initial points, the two vectors are said to be equal.

Exercise 10.2 Page No: 440

1. Compute the magnitude of the following vectors.

Solution:

Given, vectors are

2. Write two different vectors having the same magnitude.

Solution:

3. Write two different vectors having the same direction.

Solution:

Two different vectors having the same directions are

Let us

4. Find the values of x and y so that the vectors are equal

Solution:

Given vectors
will be equal only if their corresponding components are equal.

Thus, the required values of x and y are 2 and 3, respectively.

5. Find the scalar and vector components of the vector with the initial point (2, 1) and terminal point (–5, 7).

Solution:

The scalar and vector components are

The vector with initial point P (2, 1) and terminal point Q (–5, 7) can be shown as

Thus, the required scalar components are –7 and 6, while the vector components are

6. Find the sum of the vectors.

Solution:

Let us find the sum of the vectors.

7. Find the unit vector in the direction of the vector.

Solution:

We know that

8. Find the unit vector in the direction of vector, where P and Q are the points

(1, 2, 3) and (4, 5, 6), respectively.

Solution:

We know that,

9. For given vectors and , find the unit vector in the direction of the vector 

Solution:

We know that,

10. Find a vector in the direction of the vector which has magnitude of 8 units.

Solution:

Firstly,

11. Show that the vectorsare collinear.

Solution:

First,

Therefore, we can say that the given vectors are collinear.

12. Find the direction cosines of the vector 

Solution:

First,

13. Find the direction cosines of the vector joining the points A (1, 2, –3) and

B (–1, –2, 1) directed from A to B.

Solution:

We know that the given points are A (1, 2, –3) and B (–1, –2, 1).

Now,

14. Show that the vector is equally inclined to the axes OX, OY, and OZ.

Solution:

Firstly,

15. Find the position vector of a point R which divides the line joining two points P and Q, whose position vectors are , respectively, in the ratio 2:1

(i) internally

(ii) externally

Solution:

We know that

The position vector of point R dividing the line segment joining two points.

P and Q in the ratio m: n is given by

16. Find the position vector of the midpoint of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).

Solution:

The position vector of mid-point R of the vector joining points P (2, 3, 4) and Q (4, 1, – 2) is given by

17. Show that the points A, B and C with position vectors,, respectively, form the vertices of a right-angled triangle.

Solution:

We know

Given position vectors of points A, B, and C are

Hence, proved that the given points form the vertices of a right-angled triangle.

18. In triangle ABC (Fig 10.18), which of the following is not true.

Solution:

First, let us consider,

19. If are two collinear vectors, then which of the following is incorrect?

A. , for some scalar λ

B. 

C. The respective components of are proportional

D. Both the vectors have the same direction, but different magnitudes

Solution:

We know,

Exercise 10.3 Page No: 447

1. Find the angle between two vectorsandwith magnitudes √3 and 2, respectively having.

Solution:

First, let us consider,

2. Find the angle between the vectors

Solution:

Let us consider the

Hence, the angle between the vectors is cos^-1 (5/7).

3. Find the projection of the vectoron the vector.

Solution:

First,

4. Find the projection of the vectoron the vector.

Solution:

First,

Hence, the projection is 60/√114.

5. Show that each of the given three vectors is a unit vector.

Also, show that they are mutually perpendicular to each other.

Solution:

It is given that

6. Find

Solution:

Let us consider,

7. Evaluate the product

Solution:

Let us consider the given expression

8. Find the magnitude of two vectors, having the same magnitude and such that the angle between them is 60° and their scalar product is ½.

Solution:

First,

Hence, the magnitude of the two vectors is 1.

Solution:

Let us consider,

Hence, the value is √13.

10. Ifare such thatis perpendicular to, then find the value of λ.

Solution:

We know that the

11. Show that is perpendicular to, for any two nonzero vectors.

Solution:

Let us consider,

12. If, then what can be concluded about the vector?

Solution:

We know,

13. If are unit vectors such that , find the value of .

Solution:

Consider the given vectors,

Hence, the value is -3/2.

14. If either vector, then. But the converse need not be true. Justify your answer with an example.

Solution:

First,

15. If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectorsand]

Solution:

We know,

Hence, the angle is cos^-1 (10/ √102).

16. Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Solution:

Let us consider,

Given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).

Now,

Therefore, the given points A, B, and C are collinear.

17. Show that the vectorsform the vertices of a right-angled triangle.

Solution:

First, consider

Solution:

Explanation:

Exercise 10.4 Page No: 454

1. Find, if and

Solution:

It is given that,

2. Find a unit vector perpendicular to each of the vector and, where and.

Solution:

It is given that,

Solution:

First,

4. Show that

Solution:

First, consider the LHS,

We have

5. Find λ and μ if .

Solution:

It is given that

Solution:

It is given that

Solution:

First, let us consider,

9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Solution:

We know

10. Find the area of the parallelogram whose adjacent sides are determined by the vector .

Solution:

Let us consider,

Solution:

Explanation:

12. Area of a rectangle having vertices A, B, C, and D with position vectors and  , respectively is

Solution:

Explanation:

Miscellaneous Exercise Page No: 458

1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.

Solution:

Let us consider,

2. Find the scalar components and magnitude of the vector joining the points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂).

Solution:

First, let us consider,

3. A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Solution:

It is given that

Let O and B be the initial and final positions of the girl, respectively.

Then, the girl’s position can be shown as

4. If, then, is it true that? Justify your answer.

Solution:

It is given that,

5. Find the value of x for whichis a unit vector.

Solution:

We know,

6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

.

Solution:

Let us consider that the

7. If, find a unit vector parallel to the vector.

Solution:

Let us consider the given vectors,

8. Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Solution:

First, let us consider,

9. Find the position vector of a point R, which divides the line joining two points P and Q, whose position vectors areexternally in the ratio 1: 2. Also, show that P is the midpoint of the line segment RQ.

Solution:

We know,

10. The two adjacent sides of a parallelogram areand .

Find the unit vector parallel to its diagonal. Also, find its area.

Solution:

First, let us consider,

11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are.

Solution:

First,

Let’s assume a vector to be equally inclined to axes OX, OY, and OZ at angle α.

Then, the direction cosines of the vector are cos α, cos α, and cos α.

Now, we know that

Therefore, the direction cosines of the vector, which are equally inclined to the axes, are

Hence, proved.

Solution:

Assume,

13. The scalar product of the vectorwith a unit vector along the sum of vectors and is equal to one. Find the value of.

Solution:

Let’s consider the

14. If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to and.

Solution:

Let’s assume,

Hence proved.

15. Prove that, if and only if are perpendicular, given.

Solution:

It is given that

Hence, proved.

Solution:

Explanation:

Solution:

Explanation:

Hence the correct answer is D.

18. The value of is

(A) 0 (B) –1 (C) 1 (D) 3

Solution:

Explanation:

It is given that,

Hence, the correct answer is C.

Solution:

Explanation:

The major concepts of Maths covered in Chapter 10 – Vector Algebra of NCERT Solutions for Class 12 include

10.1 Introduction

10.2 Basic Concepts

• • Position Vector
  • Direction Cosines

10.3 Types of Vectors

• • Zero Vector
  • Unit Vector
  • Coinitial Vectors
  • Collinear Vectors
  • Equal Vectors
  • Negative of a Vector

10.4 Addition of Vectors

• • Properties of vector addition

10.5 Multiplication of a Vector by a Scalar

10.5.1 Components of a vector

10.5.2 Vector joining two points

10.5.3 Section formula

10.6 Product of Two Vectors

10.6.1 Scalar (or dot) product of two vectors

10.6.2 Projection of a vector on a line

10.6.3 Vector (or cross) product of two vectors

Access NCERT Solutions for Class 12 Maths Chapter 10 exercise-wise from the below links

Exercise 10.1 Solutions 5 Questions
Exercise 10.2 Solutions 19 Questions
Exercise 10.3 Solutions 18 Questions
Exercise 10.4 Solutions 12 Questions
Miscellaneous Exercise on Chapter 10 Solutions 19 Questions

NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

The chapter Vector Algebra belongs to the unit Vectors and Three-Dimensional Geometry, which adds up to 14 marks of the total marks. There are 4 exercises, along with a miscellaneous exercise in this chapter, to help students understand the concepts related to Vectors and Vector Algebra clearly. Some of the topics discussed in the tenth Chapter of NCERT Solutions for Class 12 Maths are as follows:

1. The scalar components of a vector are its direction ratios and represent its projections along the respective axes.
2. The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as l=(a/r), m=(b/r) n=(c/r)
3. The vector sum of the three sides of a triangle taken in order is 0.
4. The vector sum of two coinitial vectors is given by the diagonal of the parallelogram whose adjacent sides are the given vectors.
5. The multiplication of a given vector by a scalar λ changes the magnitude of the vector by the multiple |λ| and keeps the direction the same (or makes it opposite) accordingly, as the value of λ is positive (or negative).

The other concepts and topics explained in the NCERT Solutions of this chapter can be understood by going through Chapter 10 of the NCERT textbook for Class 12.

Key Features of NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra

Studying the Vector Algebra of Class 12 using the NCERT Solutions enables the students to understand the following: Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, the addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, the scalar triple product of vectors. Students can utilise the NCERT Solutions for Class 12 Maths Chapter 10 for any quick reference to comprehend complex topics.

Disclaimer – 

Dropped Topics – 10.7 Scalar Triple Product, 10.7.1 Coplanarity of Three Vectors