question_answer

                    Let \[\overrightarrow{a}={{a}_{1}}\hat{i}+{{a}_{2}}\hat{j}+{{a}_{3}}\hat{k},\] \[\overrightarrow{b}={{b}_{1}}\hat{i}+{{b}_{2}}\hat{j}+{{b}_{3}}\hat{k}\] and \[\overrightarrow{c}={{c}_{1}}\hat{i}+{{c}_{2}}\hat{j}+{{c}_{3}}\hat{k}\] be three non-zero vectors such that \[\overrightarrow{c}\]is unit vector perpendicular to both the vectors \[\overrightarrow{a}\]and \[\overrightarrow{b}\]. If the angle between \[\overrightarrow{a}\]and \[\overrightarrow{b}\]is \[\frac{\pi }{6},\] then \[{{\left| \begin{matrix}    {{a}_{1}} & {{a}_{2}} & {{a}_{3}}  \\    {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\    {{c}_{1}} & {{c}_{2}} & {{c}_{3}}  \\ \end{matrix} \right|}^{2}}\]is equal to :

A)  0                                

B)  1

C)  \[\frac{1}{4}\,\,(a_{1}^{2}+a_{2}^{2}+a_{3}^{3})\,\,(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})\]

D)  \[\frac{3}{4}\,\,(a_{1}^{2}+a_{2}^{2}+a_{3}^{3})\,\,(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})\,\,(c_{1}^{2}+c_{2}^{2}+c_{3}^{2})\]

E)  None of these