$$\begin{gathered} \mathrm{lf} \overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}} \mathrm{are} \mathrm{unit} \mathrm{vectors} \mathrm{such} \mathrm{that} \\ \overline{\mathrm{a}}.\overline{\mathrm{b}}=\overline{\mathrm{a}}.\overline{\mathrm{c}}=0 \mathrm{and} \left(\overline{\mathrm{b}},\overline{\mathrm{c}}\right) \frac{\mathrm{\pi }}{6}, \mathrm{then} \overline{\mathrm{a}}= \end{gathered}$$

$\begin{array}{rcl}\left|\overline{\mathrm{a}}\right|& =& \left|\overline{\mathrm{b}}\right|=\left|\overline{\mathrm{c}}\right|=1\\ \overline{\mathrm{a}}·\overline{\mathrm{b}}-\overline{\mathrm{a}}·\overline{\mathrm{c}}& =& 0 \Rightarrow \overline{\mathrm{a}}·(\overline{\mathrm{b}}-\overline{\mathrm{c}})=0\\ \left(\mathrm{or}\right) \overline{\mathrm{a}}& \perp & \overline{\mathrm{b}} \mathrm{and} \overline{\mathrm{a}}\perp \overline{\mathrm{c}} \Rightarrow \overline{\mathrm{a}} \left|\right| \overline{\mathrm{b}}\times \overline{\mathrm{c}}\\ & \Rightarrow & \overline{\mathrm{a}}=\mathrm{\lambda } (\overline{\mathrm{b}}\times \overline{\mathrm{c}})\\ \left|\overline{\mathrm{a}}\right|& =& \left|\overline{\mathrm{\lambda }}\right| |\overline{\mathrm{b}}\times \overline{\mathrm{c}}| \Rightarrow 1=\left|\mathrm{\lambda }\right| (1\times 1\times \sin \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.)\\ & \Rightarrow & \mathrm{\lambda }=\pm 2 \Rightarrow \overline{\mathrm{a}}=\pm 2(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\end{array}$