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If $\mathbf{a}=\mathbf{i}+2 \mathbf{j}-2 \mathbf{k}, \mathbf{b}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}$ and $\mathbf{c}=\mathbf{i}+3 \mathbf{j}-\mathbf{k}$, then $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})$ is equal to
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$20 \mathbf{i}-3 \mathbf{j}+7 \mathbf{k}$
$\begin{aligned} \mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\mathbf{a} \times\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -1 & 1 \\ 1 & 3 & -1\end{array}\right| & =\mathbf{a} \times(-2 \mathbf{i}+3 \mathbf{j}+7 \mathbf{k}) \\ & =\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & -2 \\ -2 & 3 & 7\end{array}\right|=20 \mathbf{i}-3 \mathbf{j}+7 \mathbf{k} .\end{aligned}$
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