= Solution
{c}
Expand the first component using the definition of the <cross product>:
$$
\begin{aligned}
[\mathbf a\times(\mathbf b\times\mathbf c)]_1
&=a_2(b_1c_2-b_2c_1)-a_3(b_3c_1-b_1c_3)\\
&=b_1(a_2c_2+a_3c_3)-c_1(a_2b_2+a_3b_3)\\
&=b_1(\mathbf a\cdot\mathbf c)-c_1(\mathbf a\cdot\mathbf b).
\end{aligned}
$$
The added $a_1b_1c_1$ terms cancel in the last step. Cyclically permuting the component indices gives the other two components, proving the <vector triple product identity>
$$
\boxed{\mathbf a\times(\mathbf b\times\mathbf c)=\mathbf b(\mathbf a\cdot\mathbf c)-\mathbf c(\mathbf a\cdot\mathbf b).}
$$
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