Find angle between A + B and A – B if A, B are vectors

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

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