Vector Algebra

Let $\bar{a} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}),\ \bar{b} = (b_1\hat{i} + b_2\hat{j} + b_3\hat{k}),\ \bar{c} = (c_1\hat{i} + c_2\hat{j} + c_3\hat{k})$ be three non-zero vectors such that $\bar{a}$ is a unit vector perpendicular to both $\bar{b}$ and $\bar{c}$. If the angle between $\bar{b}$ and $\bar{c}$ is $\dfrac{\pi}{3}$ then $\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}^2 = $