In the question its given that A, B are two vectors and we need to find out what’s angle between vectors A + B and A – B.

Simplest way to find out angle is to just take Dot Product

Dot Product of two vectors let’s say 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

So

(A + B).(A – B) = |A + B||A – B| Cosθ

Simplifying this

A.(A – B) + B.(A – B) = |A + B| |A – B| Cosθ**A.A – A.B + B.A – B.B = |A + B| |A – B| Cosθ** (Equation 1)

Dot Product of a vector with itself is just square of it’s magnitude

So if a is a vector

Then a.a = |a|^{2}

⇒ A.A = |A|^{2}

⇒ B.B = |B|^{2}

Therefore (Equation 1) can be written as

|A|^{2} – A.B + B.A – |B|^{2} = |A + B| |A – B| Cosθ

Dot Product of vectors is Commutative

Which means

A.B = B.A

|A|^{2} – A.B + A.B – |B|^{2} = |A + B| |A – B| Cosθ

|A|^{2} – |B|^{2} = |A + B| |A – B| Cosθ

θ = Cos^{-1}(|A|^{2} – |B|^{2})/(|A + B| |A – B|)

Therefore if A, B are two vectors then angle between vectors A + B and A – B is Cos^{-1}(|A|^{2} – |B|^{2})/(|A + B| |A – B|)

Where |A|, |B| are magnitudes of individual vectors and |A + B|, |A – B| are magnitudes of vectors A + B and A – B.

👇🏻 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. Dot Product of a vector with itself is square of it’s magnitude**

This can be derived from formula of Dot Product

If a, b are two vectors

Then

Their Dot Product a.b = |a| |b| Cosθ

Where

|a|, |b| are magnitudes of these vectors

θ is the angle between directions of vectors a, b

Let’s put b = a in Dot Product formula

a.a = |a| |a| Cosθ

θ = 0 (Angle between vectors a and a is zero)

a.a = |a| |a| Cos0 = |a|^{2}

a.a = |a|^{2}**3. Dot Product of two vectors is Commutative**

Which means a.b = b.a

If a, b are two vectors

This can also be proved from formula of Dot Product itself

a.b = |a| |b| Cosθ

b.a = |b| |a| Cosθ

⇒ **a.b = b.a**

Therefore Dot Product of two vectors is Commutative