What am I doing wrong here?
$(n^Tv)^2=(n^Tv)^T(n^Tv)=(v^Tn)(n^Tv)=v^T(nn^T)v=v^Tv$
$v$ is a 3d vector and $n$ is a 3d normal vector. From this calculation it looks like the projection length squared $(n^Tv)^2$ is independent of $n$, which is of course not true.