# Class 12 Maths NCERT Solutions for Chapter 10 Vector Algebra Exercise 10.3

### Vector Algebra Exercise 10.3 Solutions

1. Find the angle between two vectors a⃗ and b⃗ with magnitudes √3 and 2, respectively having a⃗ .  b⃗ = √6.

Solution

It is given that,
| a⃗ | = √3, | b⃗ | = 2 and,  a⃗ . b⃗  = √6 .

Hence, the angle between the given vectors  a⃗  and  b⃗ is Ï€/4.

2. Find the angle between the vectors  i - 2j ^  + 3k^  and  3i - 2j ^  + k^   .

Solution

The given vectors are  a = i - 2j ^  + 3k^  and  b = 3i - 2j ^  + k^  .

3. Find the projection of the vector  i - j ^  on the vector  i + j ^.

Solution

4. Find the projection of the vector  i + 3j ^  + 7k^  on the vector 7i - j ^ + 8k^ .
Solution

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

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

Thus, each of the given three vectors is a unit vector.

Hence, the given three vectors are mutually perpendicular to each other.

6. Find | a⃗ | and | b⃗ |,  if ( a⃗ +  b) . ( a⃗ - b⃗ ) = 8 and  |a⃗ | = 8| b⃗ |.
Solution

7. Evaluate the product  ( 3a⃗ - 5b⃗ ) . ( 2a⃗ +  7b⃗ ).
Solution

8. Find the magnitude of two vectors a⃗ and  b⃗ , having the same magnitude and such that the angle between them is 60° and their scalar product is  1/2 .
Solution
Let Î¸ be the angle between the vectors    a⃗ and  b⃗
It is given that  | a⃗ | = | b⃗ |,  a⃗ .  b⃗   = 1/2 , and Î¸ = 60° ...(1)
We know that  a⃗ . b⃗  = | a⃗ | | b⃗ | cos Î¸.

9. Find | x⃗ |, if for a unit vector  a⃗ , ( x⃗ -  a⃗ ).( x⃗ +  a⃗ ) = 12.
Solution

10. If  a = 2i - 2j ^  + 3k^ ,  b =  - i + 2j ^ + k^  and  c = 3i + j ^ are such that  a⃗  = Î»b⃗ is perpendicular c⃗ , then find the value of  Î» .
Solution
Hence, the required value of Î» is 8.

11. Show that (|a⃗ |b⃗  ) + (|b⃗ |a )is perpendicular to (|a⃗ |b⃗ ) - (|b⃗ |a), for any two nonzero vectors    a⃗ and  b.
Solution

Hence, | a⃗ |b + | b⃗ |a and | a⃗ |b⃗ - | b⃗ |a⃗  are perpendicular to each other.

12. If  a⃗ . a⃗ = 0 and  a⃗ . b⃗ = 0, then what can be concluded about the vector b⃗
Solution
It is given that a. a⃗ = 0 and  a⃗ . b⃗ = 0
Now,

Hence, vector b⃗ satisfying a⃗ . b⃗ = 0 can be any vector.

13. If  a⃗ , b⃗  and  c⃗  are unit vectors such that  a⃗ + b⃗  + c =  0⃗ , find the value of  a⃗ . b⃗ +  b⃗ . c⃗ +  c⃗ .  a⃗ .
Solution

14. If either vector a⃗ = 0⃗ or b⃗ =  0 then a⃗ ,  b = 0. But the converse need not be true. justify your answer with an example.
Solution

Hence, the converse of the given statement need not be true.

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 vectors and BA⃗ and BC⃗
Solution

The vertices of Î”ABC are given as A (1, 2, 3), B (–1, 0, 0), and C (0, 1, 2).
Also, it is given that ∠ABC is the angle between the vectors   BA⃗ and  BC⃗   .

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

Solution

The given point are A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) .

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

17. Show that the vectors 2i ^- j ^+k^ , i ^-3j ^-5k^  and 3i ^- 4j ^-4k^ from the vertices of a right angled triangle.

Solution

Hence, Î” ABC is a right angled triangle.

18. If  a⃗  is a nonzero vector of magnitude 'a' and Î»a⃗  is unit vector if
(A) Î» = 1
(B) Î» = -1
(C) a = |Î»|
(D) a = 1/|Î»|
Solution
Vector Î»a⃗  is a unit vector if |Î»a| = 1 .

Hence, vector Î»a⃗ is a unit vector if a = 1/|Î»|.