Consider the the following problem.
Define $u'$ and $v'$ to be the projections of $u$ and $v$ onto the plane perpendicular to the axis of $\xi$. If $\omega \in \mathbb{R}^{3 \times 1}$ is a unit vector in the direction of the axis of $\xi$, then
$u' = u -\omega \omega ^{T} u$ and $v' = v -\omega \omega ^{T} v$
Is something wrong with these equations?
I thought that $\omega \omega ^{T}u = u$ (dot product) so it becomes $u' = u - u = 0$