# If $\overrightarrow a,\:\overrightarrow b,\:\overrightarrow c$ are any three unit vectors such that $\overrightarrow a.\overrightarrow b=\overrightarrow a.\overrightarrow c=0$ and the angle between $\overrightarrow b\:\:and\:\:\overrightarrow c$ is $\large\frac{\pi}{6}$, then $\overrightarrow b\times\overrightarrow c=?$

Given: $\overrightarrow a.\overrightarrow b=\overrightarrow a.\overrightarrow c=0$
$\Rightarrow\:\overrightarrow a$ is $\perp$ to both $\overrightarrow b\:\:and\:\:\overrightarrow c$
$\Rightarrow\:\overrightarrow a=\lambda (\overrightarrow b\times\overrightarrow c)$
$\Rightarrow\:|\overrightarrow a|=|\lambda|.|\overrightarrow b\times\overrightarrow c|$
$\Rightarrow\:|\overrightarrow a|=|\lambda|.|\overrightarrow b||\overrightarrow c|. sin\large\frac{\pi}{6}$
(Since $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are unit vectors and the angle between $\overrightarrow b\:\overrightarrow c$ is $\large\frac{\pi}{6})$
$\Rightarrow\:1=|\lambda|.\large\frac{1}{2}$
or $\lambda=\pm 2$
$\Rightarrow\:\overrightarrow a=\pm (\overrightarrow b\times\overrightarrow c)$
$\Rightarrow\:\overrightarrow b\times\overrightarrow c=\pm\large\frac{1}{2}$$\overrightarrow a$