I'd like to construct a specific linear map in matrix form, and I know how to get through most of the problem, but I'm not sure how to handle the last step.
The linear map is the rotation by $\frac{5\pi}{6}$ about the main diagonal (spanned by the vector $(1,1,1)^T$), taken counterclockwise as you look toward the origin, followed by the reflection in the plane with equation $x+y+z=0$. To do the rotation part, I just need to use the Rodrigues' rotation formula, which after a little bit of work gives $ \begin{bmatrix} 1 & -\frac{\sqrt{3}+1}{2} & \frac{\sqrt{3}+1}{2} \\ \frac{\sqrt{3}+1}{2} & 1 & -\frac{\sqrt{3}+1}{2} \\ -\frac{\sqrt{3}+1}{2} & \frac{\sqrt{3}+1}{2} & 1 \end{bmatrix} $ My question now is how to handle the reflection. I don't think that it's as easy as negating the $y$ and $z$ coordinates, but that's the only thing I can think of (this thinking comes from rearranging the equation of the plane). Is there a general way to handle reflections across planes in $\mathbb{R}^3$?
However, if I wanted to instead take the orthogonal projection onto the $yz$-plane after rotating, then I would just change the $x$ coordinates to $0$ like this: $ \begin{bmatrix} 0 & 0 & 0 \\ 0& 1 & -\frac{\sqrt{3}+1}{2} \\ 0 & \frac{\sqrt{3}+1}{2} & 1 \end{bmatrix} $ Right?
I'm just trying to make sure that I really understand exactly what I'm doing.