a
(a) If two vectors are perpendicular then their dot product must be equal to zero. According to problem

\((\overrightarrow A + \overrightarrow B ).(\overrightarrow A - \overrightarrow B ) = 0\)

\(⇒\) \(\overrightarrow A .\overrightarrow A - \overrightarrow A .\overrightarrow B + \overrightarrow B .\overrightarrow A - \overrightarrow B .\overrightarrow B = 0\)

\(⇒\)\({A^2} - {B^2} = 0\)

\(⇒\) \({A^2} = {B^2}\)

\(\therefore \) \(A = B\) i.e. two vectors are equal to each other in magnitude.