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If $\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}, \boldsymbol{\beta}=-\mathbf{i}+2 \mathbf{j}-4 \mathbf{k}$ and $\mathbf{Y}=\mathbf{i}+\mathbf{j}+\mathbf{k}$, then $(\boldsymbol{\alpha} \times \boldsymbol{\beta}) \cdot(\boldsymbol{a} \times \mathbf{Y})$ is equal to
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Verified Answer
The correct answer is:
-74
$\begin{aligned}
& \mathbf{a} \cdot[\boldsymbol{\gamma} \times(\boldsymbol{\alpha} \times \boldsymbol{\beta})]=\mathbf{a} \cdot[(\boldsymbol{y} \cdot \boldsymbol{\beta}) \mathbf{a}-(\boldsymbol{y} \cdot \boldsymbol{a}) \boldsymbol{\beta}] \\
& =(\mathbf{a})^2(\boldsymbol{\gamma} \cdot \boldsymbol{\beta})-(\mathbf{y} \cdot \mathbf{a})(\boldsymbol{\beta} \cdot \mathbf{a})=14(-3)-(4)(\boldsymbol{)})=-74
\end{aligned}$
& \mathbf{a} \cdot[\boldsymbol{\gamma} \times(\boldsymbol{\alpha} \times \boldsymbol{\beta})]=\mathbf{a} \cdot[(\boldsymbol{y} \cdot \boldsymbol{\beta}) \mathbf{a}-(\boldsymbol{y} \cdot \boldsymbol{a}) \boldsymbol{\beta}] \\
& =(\mathbf{a})^2(\boldsymbol{\gamma} \cdot \boldsymbol{\beta})-(\mathbf{y} \cdot \mathbf{a})(\boldsymbol{\beta} \cdot \mathbf{a})=14(-3)-(4)(\boldsymbol{)})=-74
\end{aligned}$
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