# What is angle between a – b and a × b if a, b are vectors?

It’s given that a, b are two vectors and we need to figure out what will be angle between vectors a – b and a.b

Simplest way to solve this is to take Dot Product of vectors a – b and a.b

(a – b).(a × b)

Dot Product of two vectors A, B is defined as
A.B = |A| |B| Cosθ
Where
a, b are two vectors
|a|, |b| are magnitudes of these vectors
θ is the angle between directions of vectors a, b

Thus (a – b).(a × b) can be written as

(a – b).(a × b) = |a – b| |a × b| Cosθ
Where
θ = Angle between vectors (a – b) and (a × b)

Simplifying equation (a – b).(a × b) = |a – b| |a × b| Cosθ

a.(a × b) – b.(a × b) = |a – b| |a × b| Cosθ

Simplifying using Dot Product formula
|a| |a × b| Cos90° – |b| |a × b| Cos90° = |a – b| |a × b| Cosθ

Vector a × b is always perpendicular to plane of both vectors a and b
That’s why I took 90° angle for Dot Product

Value of Cos90° = 0

0 – 0 = |a – b| |a × b| Cosθ

0 = |a – b| |a × b| Cosθ

Cosθ = 0

General Solution for Cosθ = 0 is θ = (2n + 1)𝛑/2 where n is integer

Therefore if a, b are two vectors then possible values of angle between vectors a – b and a × b are 𝛑/2, 3𝛑/2, 5𝛑/2 and so on.

👇🏻 Key Concepts Used in This Question

1. Dot Product of two vectors a and b is defined as a.b = |a| |b| Cosθ
Where
a, b are two vectors
|a|, |b| are magnitudes of these vectors
θ is the angle between directions of vectors a, b

2. Angle between Cross Product of two vectors and vectors themselves is 90°
If a, b are two vectors
Then
a × b ⊥ a
a × b ⊥ b

3. General Solution for Cosθ = 0 is θ = (2n + 1)𝛑/2 where n is integer