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