Rotation matrix about any custom axis

Question

I know the rotation matrices about x, y and z axes but what about rotation about any other axis. I mean if I rotate whole coordinate system clockwise about axis (0,0,0) to (1,1,1) through 120 degrees, then what will be the rotation matrix?

Answer 1

The standard way to derive is

1. Select a basis $v_ x,v_y,v_z$ and write the rotation matrix $M_B$ with respect to this basis (which is the standar rotation matrix)
2. Make a change of basis from $v_ x,v_y,v_z$ to the standard basis

Answer 2

Since you know the matrices defining the rotation around the reference axes, a method to use them is the following.

You know that the action of a matrix $\bf M$ in a certain system is the same as that of the matrix $\bf M'$ in another system, when the two are related by a similarity transformation.

Then using the rotations around $x$, $y$ bring the new $z$ axis to coincide with the rotation axis, apply the required rotation around $z$, and then bring everything back to the original reference system.

Check the whole to make the necessary adjustments, depending on the convention used to build the rotation matrices.