= Solution
The <cross product> $x\times y$ is the vector perpendicular to $x$ and $y$, of magnitude $|x||y|\sin\alpha$ and orientation fixed by the right-hand rule. In <suffix notation>,
$$
(x\times y)_i=\varepsilon_{ijk}x_jy_k,
$$
where $\varepsilon_{ijk}$ is the <Levi-Civita symbol>. Therefore
$$
[x\times(x\times y)]_i
=\varepsilon_{ijk}x_j\varepsilon_{klm}x_ly_m.
$$
Using
$$
\varepsilon_{ijk}\varepsilon_{klm}
=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}
$$
gives
$$
[x\times(x\times y)]_i=x_i x_jy_j-y_i x_jx_j.
$$
Hence the <vector triple product identity> is
$$
\boxed{x\times(x\times y)=x(x\mathbin{\cdot}y)-y(x\mathbin{\cdot}x)}.
$$
Back to article page
