HOW TO FIND THE ANGLE BETWEEN TWO VECTORS

If θ is the angle between the vectors and a vector and b vector, then

Let us look into some example problems to understand finding angle between two vectors.

Question 1 :

If a vector and b vector are two vectors such that |a| = 10,|b| = 15 and a . b = 75√2, find the angle between a and b.

Solution :

θ  =  cos^-1[75√2/(10)(15)]

θ  =   cos^-1(√2/2)

θ  =   cos^-1(1/√2)

θ  =  π/4

Question 2 :

Find the angle between the vectors

(i) 2i vector + 3j vector − 6k vector and 6i vector − 3j vector + 2k vector

Solution :

Let a vector  =  2i vector + 3j vector − 6k vector and 

b vector  =  6i vector − 3j vector + 2k vector

Angle between two vector :

θ  =  cos^-1 [a vector . b vector/ |a| |b|]

a vector . b vector  =  2(6) + 3(-3) + (-6)(2)

  =  12 - 9 - 12

  =  -9

|a vector|  =  |2i vector + 3j vector − 6k vector|

r  =  √(2² + 3² + (-6)²)  =  √(4 + 9 + 36)

  =  √49  =  7

|b vector|  =  |6i vector - 3j vector + 2k vector|

r  =  √(6² + (-3)² + 2²)  =  √(36 + 9 + 4)

  =  √49  =  7

θ  =  cos^-1 [-9/(7)(7)]

θ  =  cos^-1 [-9/49]

(ii) i vector − j vector and j vector − k vector.

Let a vector  =  i vector − j vector and 

b vector  =  j vector − k vector

Solution :

Angle between two vector :

θ  =  cos^-1 [a vector . b vector/ |a| |b|]

a vector . b vector  =  1(0) + (-1)1 + 0(-1) 

  =  0 - 1 + 0

  =  -1

|a vector|  =  |i vector - j vector|

r  =  √(1² + (-1)²)  =  √(1 + 1)  =  √2

|b vector|  =  |j vector - k vector|

r  =  √(1² + (-1)²)  =  √(1+1)  =  √2

θ  =  cos^-1 [-1/( √2)( √2)]

θ  =  cos^-1 [-1/2]

θ  =  2π/3

Question 3 :

If a vector, b vector, c vector are three vectors such that a vector + 2b vector + c vector = 0 vector and |a|= 3, |b| = 4, |c| = 7,  find the angle between a vector and b vector.

Solution :

a vector + 2b vector + c vector = 0 vector

|a + 2b| vector =  |-c| vector

|a + 2b|² =  |-c|²

|a|² + |2b|² + 2|a||2b|  =  |c|²

|a|² + 4|b|² + 4|a vector| |b vector |  =  |c|²

|a|² + 4|b|² + 4(a. b)  =  |c|²

|a|² + 4|b|² + 4 |a| |b| cos θ  =  |c|²

3² + 4(4)² + 4 (3)(4) cos θ  =  7²

9 + 64 + 48 cos θ  =  49

73 + 48 cos θ  =  49

48 cos θ  =  49 - 73

48 cos θ  =  -24

θ  =  cos^-1 (-24/48)

θ  =  cos^-1 (-1/2)

θ  =  2π/3

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