Solution
= Solution
In components, part (i) gives
$$
\begin{aligned}
[a\times(b\times c)]_i
&=\epsilon_{ijk}a_j\epsilon_{klm}b_lc_m\\
&=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})
a_jb_lc_m\\
&=b_i(a\cdot c)-c_i(a\cdot b).
\end{aligned}
$$
Therefore the <vector triple product> is
$$
\boxed{a\times(b\times c)
=b(a\cdot c)-c(a\cdot b)}.
$$