If \(\bar{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=-\hat{i}+2 \hat{j}-4 \hat{k}\) and \(\bar{c}=\hat{i}+\hat{j}-2 \hat{k}\), then \((\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=\)

\(\overline{ a } \times \overline{ b }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 3 & -1 \\ -1 & 2 & -4\end{array}\right|=\hat{ i }(-10)-\hat{ j }(-9)+\hat{ k }(7)=\) \(-10 \hat{ i }+9 \hat{ j }+7 \hat{ k }\)
\(\overline{ a } \times \overline{ c }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 3 & -1 \\ 1 & 1 & -2\end{array}\right|=\hat{ i }(-5)-\hat{ j }(-3)+\hat{ k }(-1)=\) \(-5 \hat{ i }+3 \hat{ j }-\hat{ k }\)
\((\overline{\mathrm{a}} \times \overline{\mathrm{b}})(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=(-10 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}) \cdot(-5 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})=\) \(50+27-7=70\)