 # NCERT Exemplar Solutions for Class 12 Maths Chapter 10 Vector Algebra

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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.

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Exercise 10.3 Page No: 215 1. Find the unit vector in the direction of sum of vectors

Solution:

Given vectors are,  2. If find the unit vector in the direction of Solution:   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).  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: 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). 6. A vector is inclined at equal angles to the three axes. If the magnitude of is 2√3 units, find .

Solution:

As the vector 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 7. A vector has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of , given that makes an acute angle with x-axis.  8. Find a vector of magnitude 6, which is perpendicular to both the vectors and Solution:  9. Find the angle between the vectors

Solution: 10. If , show that . Interpret the result geometrically?

Solution: 11. Find the sine of the angle between the vectors and Solution: Thus, sin θ = 2/√7

12. If A, B, C, D are the points with position vectors  respectively, find the projection of along Solution:

We have, 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).  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. – Hence proved. Long 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.  16. If determine the vertices of a triangle, show that gives the

vector area of the triangle. Hence deduce the condition that the three points are collinear. Also find the unit vector normal to the plane of the triangle.

Solution:  17. Show that area of the parallelogram whose diagonals are given by and is . Also find the area of the parallelogram whose diagonals are and .

Solution:

Let’s take ABCD to be a parallelogram such that  18. If find a vector such that Solution:  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 that has magnitude 9 is Solution:

The correct option is (C). 20. The position vector of the point which divides the join of points in the ratio 3 : 1 is Solution:

The correct option is (D).

The given vectors are in the ratio 3: 1 21. The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is 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 22. The angle between two vectors and with magnitudes √3 and 4, respectively, and is Solution:

The correct option is (B). 