I was reading the following article about how to represent a $3$-D rotation in coordinates: link
I saw that in order to do a rotation around the Z-axis, the rotation matrix is: $$\begin{equation}R_Z(\theta) = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin\theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\end{equation}$$
In order to do a rotation around the X-axis, the rotation matrix is: $$\begin{equation}R_X(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \theta & -\sin \theta \\ 0 & \sin\theta & \cos \theta \end{bmatrix}.\end{equation}$$
In order to do a rotation matrix around the Y-axis, the rotation matrix is: $$\begin{equation}R_Y(\theta) = \begin{bmatrix} \cos \theta & 0 & \sin\theta \\ 0 & 1 & 0\\ -\sin \theta & 0 & \cos \theta \end{bmatrix}.\end{equation}$$
Let's say I wanted to rotate some point, $(x, y, z)$, $\alpha^{\circ}$ about the Z-axis, $\beta^{\circ}$ around the X-axis, and $\gamma^{\circ}$ around the $Y$-axis.
Then, could I represent the coordinates as: $$ \begin{equation} \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin\alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \beta & -\sin \beta \\ 0 & \sin\beta & \cos \beta \end{bmatrix} \cdot \begin{bmatrix} \cos \gamma & 0 & \sin\gamma \\ 0 & 1 & 0\\ -\sin \gamma & 0 & \cos \gamma \end{bmatrix} \cdot \begin{bmatrix}x \\ y \\ z \end{bmatrix} \end{equation}? $$
Is there some simpler way to do it so that I don't have to work with all of these matrices? Does anyone know of a general form that'll move everything into one matrix?
The Rodrigues rotation formula turns an axis and angle into a single rotation matrix, which in its most general form is an orthogonal matrix with determinant $1$. Alternatively, use quaternions.