Let a =a_ 1 i +a_ 2 j +a_ 3 k , +c_ 2 j +c_ 3 k be three non-zero vectors such that is a…

Let \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}\), \(+c_{2} \hat{j}+c_{3} \hat{k}\) be three non-zero vectors such that is a unit vector perpendicular to both the vectors and and if the angles between and is \[\frac{\pi}{6} \ ext {, then }\left|\begin{array}{lll} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right|^{2}\] is equal to