$\overrightarrow{\mathrm{P}}=$ vector $\mathrm{sum}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$

$\overrightarrow{\mathrm{Q}}=$ vector differences $=\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}$

Since $\overrightarrow{\mathrm{P}}$ and $\overrightarrow{\mathrm{Q}}$ are perpendicular

$\therefore \overrightarrow{\mathrm{P}} \cdot \overrightarrow{\mathrm{Q}}=0$

$\Rightarrow(\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}) \cdot(\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}})=0 \Rightarrow \mathrm{A}^{2}=\mathrm{B}^{2}$

$\Rightarrow|\mathrm{A}|=|\mathrm{B}|$