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# If $\overrightarrow a.\overrightarrow b=\overrightarrow a.\overrightarrow c$ and $\overrightarrow a\times\overrightarrow b=\overrightarrow a\times\overrightarrow c$ and if $\overrightarrow a\neq 0$, then

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Given :$\overrightarrow a.\overrightarrow b=\overrightarrow a.\overrightarrow c$ and $\overrightarrow a\times\overrightarrow b=\overrightarrow a\times\overrightarrow c$
$\Rightarrow\:\overrightarrow a.(\overrightarrow b-\overrightarrow c)$ and $\overrightarrow a\times (\overrightarrow b-\overrightarrow c)$
$\Rightarrow\:\overrightarrow a=0\:\:or\:\:\overrightarrow b-\overrightarrow c=\overrightarrow 0\:\: or\:\:\overrightarrow a$ is $\perp$ and $||$ to $\overrightarrow b-\overrightarrow c$
But since given that $\overrightarrow a\neq \overrightarrow 0$,and also since
$\overrightarrow a$ cannot be $\perp$ and $||$ to $\overrightarrow b-\overrightarrow c$, $\overrightarrow b-\overrightarrow c=\overrightarrow 0$
$i.e.,\:\:\overrightarrow b=\overrightarrow c$
answered Nov 21, 2013