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 .
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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 :
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Also, show that they are mutually perpendicular to each other.Solution
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Thus, each of the given three vectors is a unit vector.
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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
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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
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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 .
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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) .
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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
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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 .
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Hence, vector λa
⃗ is a unit vector if a = 1/|λ|.
The correct answer is D.
