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Q.
The sum of two vectors $\vec{A}$ and $\vec{B}$ is at right angles to their difference. Then
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Solution:
The sum of two vectors is $\vec{R}=\vec{A}+\vec{B}$
The difference of two vectors is $\vec{R}=\vec{A}-\vec{B}$
Since $\vec{R}$ and $\vec{R}$ are at right angles, therefore their dot product is zero
i.e., $\vec{R}\cdot\vec{R}=0$ or, $\left(\vec{A}+\vec{B}\right)\cdot\left(\vec{A}-\vec{B}\right)=0$
or $\vec{A}\cdot\vec{A}-\vec{A}\cdot\vec{B}+\vec{B}\cdot\vec{A}-\vec{B}\cdot\vec{B}=0$
or,$ A^{2}-\vec{A}\cdot\vec{B}+\vec{A}\cdot\vec{B}-B^{2}=0$
(since $\vec{A}\cdot\vec{B}=\vec{B}\cdot\vec{A})$
$\therefore A^{2}-B^{2}=0$
$\Rightarrow A=B$