When rotating reference frames using the XYZ fixed angle convention, why do we multiply matrices in the inverse order that rotation was done?

Question

I'm reading about the XYZ fixed angle convention for relating reference frames. It says that to solve for the description of frame {B} in frame {A} we do the following:

Start with the frame coincident with a known reference frame {A}. Rotate {B} first about $X_A$ by an angle $\gamma$ , then about $Y_A$ by an angle $\beta$, and, finally, about $Z_A$ by an angle $\alpha$.

Conceptually, why do we multiply in the order $R_Z(\alpha)R_Y(\beta)R_X(\gamma)$ when the actual rotation occurred in the reverse order?

Drawing of each step in the process

Answer 1

It depends on your conventions on how a matrix $A$ acts on a vector $v$. Does the matrix act on the left ($Av$) or the right ($vA$)? This is essentially a question of whether you think of the vectors as row vectors or column vectors.

If the matrix acts on the left, then $AB$ would act as $ABv$. As you can see, with this convention, $B$ acts on $v$ first, then $A$. So $AB$ means do $B$, then $A$.

If the matrix acts on the right, then $AB$ would act as $vAB$. With this alternate convention, $A$ acts first, then $B$. So $AB$ means do $A$, then $B$.

Either convention is fine. But you have to be consistent throughout, or the calculations will be wrong.