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# If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are any three vectors such that $\overrightarrow a=\overrightarrow b+\overrightarrow c$ and $\overrightarrow b$ is $\perp$ to $\overrightarrow c$, then ?

$\begin{array}{1 1} (a)\:\:a^2=b^2+c^2\:\:\qquad\:\:(b)\:\:b^2=c^2+a^2\:\:\qquad\:\:(c)\:\:c^2=a^2+b^2\:\:\qquad\:\:(d)\:\:2a^2-b^2=c^2 \end{array}$

Given : $\overrightarrow a=\overrightarrow b+\overrightarrow c$
$\Rightarrow\:|\overrightarrow a|^2=a^2=|\overrightarrow b+\overrightarrow c|^2$
$\Rightarrow\:a^2=b^2+c^2+2\overrightarrow b.\overrightarrow c$
But also it is given that $\overrightarrow b$ is $\perp$ to $\overrightarrow c$
$\Rightarrow\:\overrightarrow b.\overrightarrow c=0$
$\therefore\:a^2=b^2+c^2$