Rotating points with horizontal and vertical tilt

Question

From Wikipedia, I've seen that if I have a rotation to do in three dimensions, it must be around an axis in order to do so. However, I have a rotation along the z-axis along with the xy-plane (aka a horizontal and vertical tilt respectively). How might I condense the calculation into one matrix?

Answer 1

A rotation with angle $\theta$ around the origin in the $(O,x,y)$ plane has matrix: $$\begin{bmatrix}\cos\theta& -\sin\theta\\\sin\theta &\cos\theta \end{bmatrix}$$ hence the matrix of a rotation of angle $\theta$ around the $z$ axis in $3$-space has matrix: $$\begin{bmatrix}\cos\theta& -\sin\theta &0 \\\sin\theta &\cos\theta& 0 \\ 0&0&1\end{bmatrix}$$