@dvernet No, and the reason is given by the two illustrations on the right (note the z direction). And note that the signs of the components in the transformation matrix also depend on the coordinate system you are working (left-handed or right-handed).

Should $R_{y, \theta}$ be: $$ R_{y, \theta} = \begin{bmatrix} \cos{\theta} & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & \cos(\theta) \end{bmatrix} $$

@dvernet No, and the reason is given by the two illustrations on the right (note the z direction). And note that the signs of the components in the transformation matrix also depend on the coordinate system you are working (left-handed or right-handed).

Ahh, thanks @skygao