If $\overrightarrow a, \overrightarrow b$ and $\overrightarrow c$ are three unit vectors such that $\overrightarrow a.\overrightarrow b = \overrightarrow a. \overrightarrow c = 0$ and angle between $\overrightarrow b \: and \: \overrightarrow c$ is $\large\frac{\pi}{6},$ prove that $\overrightarrow a = \pm\: 2 ( \overrightarrow b$ x $\overrightarrow c )$.

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