How should the order of application of rotation transformation be interpreted (in PowerPoint)?

Question

For a university assignment, I have a question about rotating a picture in PowerPoint. In PowerPoint, a picture can have four transformations. Rotate right (90°), rotate left (90°), flip horizontally and flip vertically. Let's call them $R(r)$, $R(l)$, $F(v)$ and $F(h)$ for short.

We are then asked to compute the matrix multiplication for every pair of possible transformations. For example, $R(r)*R(r)$ or $R(r)*F(h)$.

All of that seems fairly simple to me, but they specified something in the question I don't understand.

$R(r)*R(l)$ is read from right to left as "rotate left" then "rotate the result right", because Powerpoint transformations are performed in world coordinates.

Can someone explain to me why I would read that left to right? And does that mean when I multiply the matrices together I would, in fact, do $R(l)*R(r)$ when asked to do $R(r)*R(l)$?

Why would world co-ordinates effect this?

Would I read all transformations as left to right or just some?

Answer 1

The use of “world coordinates,” whatever that might mean in the context of PowerPoint, doesn’t really have anything to do with why the transformation matrices are applied right to left. If you write the coordinates of a point as a column vector, then you left-multiply it by a matrix to get the transformed point. If you want to chain another transformation onto that, you put it on the left side, so the chains build right-to-left. If you write the coordinates of a point as a row vector, on the other hand, then you multiply on the right by the transformation matrices, and chains of transformations build left-to-right.

Answer 2

The product $AB$ of matrices is the effect of applying $A$ after applying $B$, simply because we write the "effect" of a matrix $A$ on a vector $v$ as $Av$ (i.e., the convention is to work with column vectors). Thus $(AB)v=A(Bv)$ is $A$ applied to $Bv$. Note that this convention matches the usual notation for (nested) functions: $\log(\sin(x))$ is the logarithm function applied to the result from applying the sine function to $x$.

Of course, the rules for computing the product of matrices are just right for this convention. Or, if one wanted $AB$ to stand for "first apply $A$, then apply $B$", one would use row vectors instead and write "$A$ aplied to $v$" as $vA$ - which is very uncommon (and in effect just means that one works with the transposed matrices, relative to the usual convention).