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

It’s given that |A + B| = |A| – |B| and A, B are two vectors
We need to find out what’s angle between vectors A, B

Let’s start with what’s given in the question
|A + B| = |A| – |B|

Squaring both sides of this equation
|A + B|2 = (|A| – |B|)2

We know that Dot Product of a vector with itself is square of it’s magnitude
Which means
a.a = |a|2
Where
a is a vector

Using this fact to further simplify above equation
(A + B).(A + B) = (|A| – |B|)2

A.(A + B) + B.(A + B) = (|A| – |B|)2

A.A + A.B + B.A + B.B = (|A| – |B|)2

Right Hand Side of above equation is just a number because both |A| and |B| are numbers so square of their difference will also be a number.

Expanding right hand side using formula (a – b)2 = a2 + b2 + 2ab (Where a, b are just numbers)

A.A + A.B + B.A + B.B = |A|2 + |B|2 + 2|A| |B|

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

⇒ A.A = |A|2
⇒ B.B = |B|2

|A|2 + A.B + B.A + |B|2 = |A|2 + |B|2 + 2 |A| |B|

Also Dot Product of two vectors is Commutative which means
a.b = b.a
Where
a, b are two vectors

⇒ A.B = B.A

|A|2 + A.B + A.B + |B|2 = |A|2 + |B|2 + 2 |A| |B|

Simplifying

A.B + A.B = 2 |A| |B|

2 A.B = 2 |A| |B|

A.B = |A| |B|

Using Dot Product Formula
|A| |B| Cosθ = |A| |B|

Cosθ = 1

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

Therefore if |A + B| = |A| – |B| then possible values of angle between vectors A, B are 0, 2𝛑, 4𝛑, 6𝛑 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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