Rotation around a line

Question

Let $F : \mathbb{E}^3 \to \mathbb{E}^3$ be one of the possible orthogonal rotations of angle $5/6 \pi$ around the line $r$ of equations $y-5z + 2 =0$ and $4x-y-10z = 0$.

Determine the matrix associated to $F$ with respect to the canonical basis. Is such a rotation necessarily a direct isometry?

Without explaining the whole solution, what are the main steps of such a problem? Thanks.

Answer 1

I would break the computation into a few steps.

1. Compute the $3 \times 3$ matrix $M$ associated to the rotation of angle $5/6\pi$ around the $z$-axis.
2. Use the given equations to compute the unit column vector $u_r$ parallel to the line $r$.
3. Compute an orthonormal matrix $R$ which takes the unit column vector $u_z = \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix}$ to $u_r$, i.e. $R u_z = u_r$.
4. Compute $RMR^{-1}$.