NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra

The NCERT Exemplar textbooks are very important for students in order to attain strong conceptual knowledge. Students can now access the NCERT Exemplar Solutions available subject-wise for getting guidance to solve the exemplar problems. All the solutions are created to help students at various levels of their difficulty. Further, all solutions are created by subject matter experts, following the latest CBSE patterns.

The 10th Chapter of NCERT Exemplar Solutions for Class 12 Mathematics is Vector Algebra. Here, students will learn introduction to vectors, its types, addition and multiplication performed on vectors, components of the vector, section formula, scalar product, projection of a vector on a line and vector product. To attain a strong grip over the concepts of this chapter, students can make use of the solutions PDF of NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra, from the link given below.

Download the PDF of NCERT Exemplar Solutions For Class 12 Maths Chapter 10 Vector Algebra

 

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Access Answers to NCERT Exemplar Class 12 Maths Chapter 10 Vector Algebra

Exercise 10.3 Page No: 215

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 1

1. Find the unit vector in the direction of sum of vectors

Solution:

Given vectors are,

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 2

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 3

2. If find the unit vector in the direction of

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 4

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 5

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 6

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 7

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

Solution:

Given coordinates are P(5, 0, 8) and Q(3, 3, 2).

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 8

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 9

4. If 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.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 10

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

Solution:

Let the given points be A(k, – 10, 3), B(1, –1, 3) and C(3, 5, 3).

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 11

6. A vector NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 12is inclined at equal angles to the three axes. If the magnitude of NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 13is 2√3 units, findNCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 14.

Solution:

As the vector NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 15
makes equal angles with the axes, their direction cosines should also be same

So, l = m = n

And we know that,

l2 + m2 + n2 = 1 ⇒ l2 + l2 + l2 = 1

3l2 = 1

l = ± 1/√3

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 16

7. A vector NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 17has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components ofNCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 18 , given thatNCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 19 makes an acute angle with x-axis.

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 20

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 21

8. Find a vector of magnitude 6, which is perpendicular to both the vectors NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 22and NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 23

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 24

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 25

9. Find the angle between the vectors

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 26

10. If NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 27, show that NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 28. Interpret the result geometrically?

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 29

11. Find the sine of the angle between the vectors NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 30and NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 31

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 32

Thus, sin θ = 2/√7

12. If A, B, C, D are the points with position vectors NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 33NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 34respectively, find the projection of NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 35alongNCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 36

Solution:

We have,

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 37

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

Solution:

Given vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 38

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 39

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

Solution:

Let’s consider ABCD and ABFE be two parallelograms on the same base AB and between same parallel lines AB and DF.

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 40

– Hence proved.

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 41Long Answer (L.A.)

15. Prove that in any triangle ABC, , where a, b, c are the

magnitudes of the sides opposite to the vertices A, B, C, respectively.

Solution:

In triangle ABC, the components of c are c cos A and c sin A.

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 42

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 4316. If NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 44determine the vertices of a triangle, show that gives the

vector area of the triangle. Hence deduce the condition that the three points NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 45 are collinear. Also find the unit vector normal to the plane of the triangle.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 46

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 4717. Show that area of the parallelogram whose diagonals are given by NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 48 and NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 49 is . Also find the area of the parallelogram whose diagonals are NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 50and NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 51 .

Solution:

Let’s take ABCD to be a parallelogram such that

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 52

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 53

18. If NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 54 find a vector NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 55such that NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 56

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 57

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 58

Objective Type Questions

Choose the correct answer from the given four options in each of the Exercises from 19 to 33 (M.C.Q)

19. The vector in the direction of the vector NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 59 that has magnitude 9 is

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 60

Solution:

The correct option is (C).

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 61

20. The position vector of the point which divides the join of points NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 62 in the ratio 3 : 1 is

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 63

Solution:

The correct option is (D).

The given vectors are in the ratio 3: 1

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 64

21. The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 65

Solution:

The correct option is (C).

Let A and B be two points whose coordinates are given as (2, 5, 0) and (-3, 7, 4)

So, we have

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 66

22. The angle between two vectors NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 67and NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 68with magnitudes √3 and 4, respectively, and NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 69is

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 70

Solution:

The correct option is (B).

NCERT Exemplar Solutions Class 12 Mathematics Chapter 10 - 71

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