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$\overrightarrow{\mathbf{a}} \times[\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})]$ is equal to
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The correct answer is:
$(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})$
$\begin{aligned} \overrightarrow{\mathbf{a}} \times[\overrightarrow{\mathbf{a}}&\times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})] \\ &=\overrightarrow{\mathbf{a}} \times\{(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) \overrightarrow{\mathbf{a}}-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}}) \overrightarrow{\mathbf{b}}\} \\ &=(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}})-(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \\ &=(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}) 0+(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}) \\ &=(\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{a}})(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}) \end{aligned}$
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