Unit vector rotated by 30° in clockwise direction with respect to specific axis of rotation

Question

I have a vector from point (0,0,0) to (1,1,1) that specifies my axis of rotation. The vector being rotated is the unit vector along y-axis. The rotation is clockwise when looking from the positive side of the axis of rotation.
How to find the resulting vector after rotation?

Answer 1

There are two steps:

1. Define the rotation: Your choices are (a) the Rodrigues' rotation formula to get a rotation matrix or (b) the quaternions to get a unit quaternion representing the rotation. In either case your axis would be $[1, 1, 1]/\sqrt 3$ and your angle would be $-30^{\circ}$ (Note for quaternions you would use half the angle but that should show up in the formula). In either case you would get a rotation matrix $\mathbf R$ or a unit quaternion $\mathbf q$.

2. Rotate the stated vector using the chosen method. For the rotation matrix case your vector would be $\mathbf v = [0, 1, 0]$ which would give you $\mathbf{Rv}$. For the quaternion case the vector would be $\mathbf u= [0, 0, 1, 0] = [0, \mathbf v]$ (note that 0 was prepended to $\mathbf v$ to make it a quaternion. The convention I am using is that the real part is the first element). Then the rotation would be $\mathbf{qu}\bar{\mathbf q}$ where $\bar{\mathbf q}$ is the conjugate of $\mathbf{q}$. Your resulting vector would be the last three elements of the result.

This is an overview of the process allowing you to fill in the blanks. Is there a particular aspect that you are struggling with?