If three non-zero vectors are \[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k},\] \[\mathbf{b}={{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\] and \[\mathbf{c}={{c}_{1}}\mathbf{i}+{{c}_{2}}\mathbf{j}+{{c}_{3}}\mathbf{k}.\] If c is the unit vector perpendicular to the vectors a and b and the angle between a and 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 [IIT 1986]

• A. 0
• B. \[\frac{3\,(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})\,(\Sigma c_{1}^{2})}{4}\]
• C. 1
• D. \[\frac{(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})}{4}\]