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# If $\overrightarrow b\:\:is\:\: \perp\:\:to \:\:\overrightarrow a+\overrightarrow b\:\:and\:\:\overrightarrow a\:\:is\:\:\perp \:\:to\:\:2\overrightarrow b+\overrightarrow a$ then $|\overrightarrow a|=?$

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Given: $\overrightarrow b$ is $\perp$ to $\overrightarrow a+\overrightarrow b$ and $\overrightarrow a$ is $\perp$ to $2\overrightarrow b+\overrightarrow a$
$\Rightarrow\:\overrightarrow b.(\overrightarrow a+\overrightarrow b)=0\:\:and\:\:\overrightarrow a.(2\overrightarrow b+\overrightarrow a)=0$
$\Rightarrow\:\overrightarrow a.\overrightarrow b+|\overrightarrow b|^2=0\:\:and\:\:2\overrightarrow a.\overrightarrow b+|\overrightarrow a|^2=0$
$\Rightarrow\:|\overrightarrow a|^2=2|\overrightarrow b|^2$
or $|\overrightarrow a|=\sqrt 2 |\overrightarrow b|$
answered Dec 8, 2013