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The vectors $\vec{A}$ and $\vec{B}$ are such that $|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|$. The angle between the two vectors is :-
A
$90^\circ$
B
$60^\circ$
C
$75^\circ$
D
$45^\circ$
Detailed Solution
$|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}| \Rightarrow |\vec{A} + \vec{B}|^2 = |\vec{A} - \vec{B}|^2$
$|\vec{A} + \vec{B}|^2 = A^2 + B^2 + 2AB\cos\theta$
$|\vec{A} - \vec{B}|^2 = A^2 + B^2 - 2AB\cos\theta$
$\Rightarrow A^2 + B^2 + 2AB\cos\theta = A^2 + B^2 - 2AB\cos\theta$
$\Rightarrow 4AB\cos\theta = 0 \Rightarrow \cos\theta = 0$
$\Rightarrow \theta = 90^\circ$
$|\vec{A} + \vec{B}|^2 = A^2 + B^2 + 2AB\cos\theta$
$|\vec{A} - \vec{B}|^2 = A^2 + B^2 - 2AB\cos\theta$
$\Rightarrow A^2 + B^2 + 2AB\cos\theta = A^2 + B^2 - 2AB\cos\theta$
$\Rightarrow 4AB\cos\theta = 0 \Rightarrow \cos\theta = 0$
$\Rightarrow \theta = 90^\circ$