If |A + B| = |A – B| and A,B are vectors then find out angle between vector A and B

In this question, we’re given that |A + B| = |A – B| and A, B are vectors
What we’re trying to find out is angle between vectors A and B

So let’s start with what’s given |A + B| = |A – B|

Squaring both sides of this equation

|A + B|2 = |A – B|2 (Equation 1)

Dot Product of a vector with itself is square of it’s magnitude
This means if a is a vector
Then
|a|2 = a.a

Using this formula
(Equation 1) can be written as

(A + B).(A + B) = (A – B).(A – B)

Simplifying
A.(A + B) + B.(A + B) = A.(A – B) – B.(A – B)

A.A + A.B + B.A + B.B = A.A – A.B – B.A + B.B

Cancelling out values on both sides
A.B + B.A = – A.B – B.A (Equation 2)

Dot Product of vectors is Commutative
Which means
If a, b are two vectors
Then a.b = b.a

Using this formula (Equation 2) can be written as
A.B + A.B = – A.B – A.B

2 A.B = – 2 A.B

2 A.B + 2 A.B = 0

4 A.B = 0

A.B = 0

|A| |B| Cosθ = 0

⇒ Cosθ = 0

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

Therefore if |A + B| = |A – B| and A, B are two vectors then possible values of angles between vectors A and B are 𝛑/2, 3𝛑/2, 5𝛑/2 and so on.

👇🏻 Key Concepts Used in This Question

1. 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

2. 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

3. General Solution for Cosθ = 1 is θ = 2n𝛑 where n is integer

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