Multiplying two rotation matrices I have a set of 3 Euler angles which I have converted into a rotation matrix (R_in) in the ZYZ convention. I now want to apply a rotation onto these euler angles using a pre-defined rotation matrix (R_mult). What is the right way to perform the rotation?
1. R_rotated = R_in · R_mult
2. R_rotated = R_mult · R_in

3. R_rotated = R_in · transpose(R_mult)
4. R_rotated = transpose(R_mult) · R_in

5. R_rotated = transpose(R_in) · R_mult
6. R_rotated = R_mult · transpose(R_in)

where "·" implies a dot product.

Also for completeness of understanding, what are the physical implications of the wrong options?
 A: The correct order is $R_{\mathrm{mult}}R_{\mathrm{in}}$.
For two rotations $R_1,R_2$, the product $R_1 \cdot R_2$ is the matrix corresponding to the rotation obtained by first applying $R_2$, then applying $R_1$. That is, rotations are done "from right to left". In our case, the first rotation (corresponding to $R_{\mathrm{in}}$) is the rotation that moves the $x,y,z$ axes to their new orientations (corresponding to the Euler angles), and the second rotation is $R_{\mathrm{mult}}$ is the pre-defined rotation (applied to our appropriately pre-rotated space).
Multiplication by $R^T$ for a rotation matrix $R$ corresponds to the inverse of a rotation $R$, i.e. undoing the rotation in question.
For example, if we wanted to find the matrix of the rotation matrix $R$ relative to the coordinates defined by the axes corresponding to our Euler angles, then the corresponding matrix would be computed as $R^T_{\mathrm{in}} R_{\mathrm{mult}} R_{\mathrm{in}}$. In other words, to rotate something in this new coordinate system, we could first apply $R_{\mathrm{in}}$ (convert alternative coordinates to the standard coordinates), apply $R_{\mathrm{mult}}$, then undo $R_{\mathrm{in}}$ (convert the standard-coordinate result back to the alternative coordinates).
