= Solution
Using the <Levi-Civita symbol> and the <contraction of two Levi-Civita symbols>, the $i$th component of the <vector triple product> is
$$
\epsilon_{ijk}a_j\epsilon_{klm}b_lc_m
=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})a_jb_lc_m
=b_i(a_jc_j)-c_i(a_jb_j).
$$
Thus \b[the triple-product identity is] $a\times(b\times c)=(a\cdot c)b-(a\cdot b)c$.
Apply this with the unit vector $n$ to obtain $\Pi(x)=x-(n\cdot x)n$. <Dot products> and scalar multiplication are linear in $x$, so $\Pi(sx+ty)=s\Pi(x)+t\Pi(y)$. Its <matrix representation> is
$$
\boxed{P=I-nn^T,\qquad P_{ij}=\delta_{ij}-n_in_j.}
$$
The fixed-vector equation is equivalent to $(n\cdot x)n=0$. Since $n\ne0$, \b[all its solutions are] $x\in n^\perp$. Also $\boxed{\Pi(n)=0}$. Every vector decomposes as $x=\Pi(x)+(n\cdot x)n$, with $\Pi(x)$ orthogonal to $n$. Hence $\Pi$ is the <orthogonal projection> onto the plane through the origin normal to $n$: it fixes that plane and removes the perpendicular component.
Back to article page
