Matrix multiplications are ``backwards''

Which operation is getting applied to the model first when we apply a product of matrices? Consider rotating a point $p = (x,y,z)$. We have two rotation matrices $R$ and $Q$. If we rotate $p$ using $R$, we obtain $p' = Rp$. If we then apply $Q$, we get $p'' = Qp'$. Now suppose that we instead want to first combine the two rotations and then apply them to $p$ to get $p''$. Programmers are often temped to combine them as $RQ$ because we read from left to right and also write sequences in this way. However, it is backwards for linear algebra because $Rp$ is already acting from the left side. Thus, it ``reads'' from right to left.^3.1 We therefore must combine the rotations as $QR$ to obtain $p'' = QRp$. Later in this chapter, we will be chaining together several matrix transforms. Read them from right to left to understand what they are doing!