The vectors $\vec{A}$ and $\vec{B}$ are such that $|\vec{A}+\vec{B}|=|\vec{A}-\vec{B}|$. The angle between…

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:
  1. $45^{\circ}$
  2. $90^{\circ}$
  3. $60^{\circ}$
  4. $75^{\circ}$

Solution

According to the question:
$\begin{aligned}
|\vec{A}+\vec{B}| & =|\vec{A}-\vec{B}| \\
\Rightarrow |\vec{A}+\vec{B}|^2 & =|\vec{A}-\vec{B}|^2
\end{aligned}$
$\begin{aligned}
& A^2+B^2+2 A B=A^2+B^2-2 A B \\
& 4 \vec{A} \cdot \vec{B}=0 \\
& \vec{A} \cdot \vec{B}=0 \\
& \Rightarrow A B \cos \theta=0 \\
& \Rightarrow \theta=90^{\circ} \\
& {[\text {As } A=B \ne 0] }
\end{aligned}$

Asked in: NEET 2006

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