NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra is provided below. Students can easily practice and learn about the difference between a scalar and a vector quantity, their properties, operations of vectors, etc. The chapter has a significant role in helping students score high marks not only in board exams but also in competitive exams. The NCERT Solutions for Class 12 Maths Vector Algebra are extremely crucial for the preparation of the various problems asked during the Class 12 Mathematics board examination.

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra Exercise 10.1 is based on the following topics:

1. Introduction
2. Basic Concepts
   • Position Vector
   • Direction Cosines

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra Exercise 10.2 is based on the following topics:

1. Addition of Vectors
   1. Properties of vector addition
2. Multiplication of a Vector by a Scalar
   1. Components of a vector
   2. Vector joining two points
   3. Section formula

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3 is based on the following topics:

1. Product of Two Vectors
   1. Scalar (or dot) product of two vectors: The result of a scalar product of two vectors is a scalar quantity. Two vectors, with magnitudes not equal to zero, are perpendicular only if their scalar product is equal to zero.
   2. Projection of a vector on a line

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra Exercise 10.4 is based on the vector (or cross) product of two vectors

 

NCERT Solutions Class 12 Maths Chapter 10 Vector Algebra Algebra Exercise 10.1

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

Solution:

The vector

represents the displacement of 40 km, 30o 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 metres 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.

NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra 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,

NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra 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:

NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra 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:

NCERT Solutions for Class 12 Maths Chapter 10 – Vector Algebra 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: