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 π/4.
2. Find the angle between the vectors i ^ - 2j ^ + 3k^ and 3i ^ - 2j ^ + k^ .
Solution
The given vectors are a⃗ = i ^ - 2j ^ + 3k^ and b⃗ = 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 + b⃗) . ( 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 θ be the angle between the vectors a⃗ and b⃗ .
It is given that | a⃗ | = | b⃗ |, a⃗ . b⃗ = 1/2 , and θ = 60° ...(1)
We know that a⃗ . b⃗ = | a⃗ | | b⃗ | cos θ.
9. Find | x⃗ |, if for a unit vector a⃗ , ( x⃗ - a⃗ ).( x⃗ + a⃗ ) = 12.
Solution
10. If a⃗ = 2i ^ - 2j ^ + 3k^ , b⃗ = - i ^ + 2j ^ + k^ and c⃗ = 3i ^ + j ^ are such that a⃗ = λb⃗ is perpendicular c⃗ , then find the value of λ .
Solution
Hence, the required value of λ is 8.
11. Show that (|a⃗ |b⃗ ) + (|b⃗ |a⃗ )is perpendicular to (|a⃗ |b⃗ ) - (|b⃗ |a⃗), for any two nonzero vectors a⃗ and b⃗ .
Solution
Hence, | a
⃗ |b
⃗ + | b
⃗ |a
⃗ and | a
⃗ |b
⃗ - | b
⃗ |a
⃗ 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 a⃗. 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 + c⃗ = 0⃗ , find the value of a⃗ . b⃗ + b⃗ . c⃗ + c⃗ . a⃗ .
Solution
14. If either vector a⃗ = 0⃗ or b⃗ = 0⃗ then a⃗ , b⃗ = 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 ΔABC 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, Δ ABC is a right angled triangle.
18. If a⃗ is a nonzero vector of magnitude 'a' and λa⃗ is unit vector if
(A) λ = 1
(B) λ = -1
(C) a = |λ|
(D) a = 1/|λ|
Solution
Vector λa⃗ is a unit vector if |λa⃗| = 1 .
Hence, vector λa
⃗ is a unit vector if a = 1/|λ|.
The correct answer is D.
