How do you find the composition $F_1 \circ F_2 \circ F_3$ of three affine transformations if $F_1$ is the reflection about the $yz$-plane, $F_3$ is the rotation over $\pi /2$ around the x-axis and $F_2$ is the translation in the direction of $(1,1,1)$?
It's easy enough to find $F_1$ and $F_3$, and compose them, because they're purely linear (so it's just matrix multiplication), but I don't know what to do with the linear part $F_2$?
If $A,B,C$ are the linear parts of $F_1,F_2,F_3$ respectively, and $F_1 = A$, $F_2= t_b \circ B, F_3 = C$, then for any $p \in \mathbb{E}^3$: $$ F_1 \circ F_2 \circ F_3(p) = ABC(p ) + Ab + b.$$