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**