If vectors |A + B| = |A - B| then what can be the angle between A and B ?

Let θ be the angle between \(\overset\rightarrow{A}\) and \(\overset\rightarrow{B}\), then

\(|\overset\rightarrow{A} + \overset\rightarrow{B}|^2\) = A² + B² + 2AB cosθ

Also the angle between \(\overset\rightarrow{A}\) and \(-\,\overset\rightarrow{B}\) is (180° - θ)

Hence, \(|\overset\rightarrow{A} + \overset\rightarrow{B}|^2\) = A² + B² + 2AB cos(180° - θ)

= A² + B² - 2AB cosθ

As \(|\overset\rightarrow{A} + \overset\rightarrow{B}|\) = \(|\overset\rightarrow{A} - \overset\rightarrow{B}|\), we can equate above two equations, 2AB cosθ

\(\Rightarrow \) 4AB cosθ = 0

Assuming \(\overset\rightarrow{A}\) and \(\,\overset\rightarrow{B}\) as non-zero vector,

we get, cosθ = 0

\(\Rightarrow \) θ = 90°

Thus, if |\(\overset\rightarrow{A}\) + \(\overset\rightarrow{A}\)| = |\(\overset\rightarrow{A}\) - \(\overset\rightarrow{B}\)|, then vectors \(\overset\rightarrow{A}\) and \(\overset\rightarrow{B}\) must be at right angles to each other.