In which order can you execute the given rotation and the projection successively?

Question

Given is the rotation $$d:\mathbb{R}^2 \rightarrow \mathbb{R}^2 \mbox{ with } d:\begin{pmatrix} x\\ y \end{pmatrix} \mapsto \begin{pmatrix} x \cos \alpha - y \sin \alpha\\ x \sin \alpha + y \cos \alpha \end{pmatrix}$$

and the projection

$$p: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \mbox{ with } p: \begin{pmatrix} x\\ y\\ z \end{pmatrix} \mapsto \begin{pmatrix} x\\ y \end{pmatrix}$$

In which order can you execute the rotation $d$ and projection $p$ successively?

I'm not quite sure why the order would matter. So if we form the linear mapping to a matrix, we have for rotation:

$$\begin{pmatrix} \cos \alpha & -\sin \alpha\\ \sin \alpha & \cos \alpha \end{pmatrix}$$

And for the projection we can take the matrix

$$\begin{pmatrix} 1\\ 0 \end{pmatrix}$$

because projection is linear mapping from vector space to itself where its square is still the same result.

And now multiply both matrices?

I don't see why the order matters here and how to do it actually? Does the order matter because if you choose it badly you cannot do matrix multiplication because their sizes don't match?

Answer 1

Since

• $d:\mathbb{R}^2 \rightarrow \mathbb{R}^2$

and

• $p: \mathbb{R}^3 \rightarrow \mathbb{R}^2$

the composition is possible only for projection first and then rotation, that is

• $d\circ p: \mathbb{R}^3 \rightarrow \mathbb{R}^2\quad \begin{pmatrix} x\\ y\\ z \end{pmatrix} \mapsto \begin{pmatrix} x \cos \alpha - y \sin \alpha\\ x \sin \alpha + y \cos \alpha \end{pmatrix}$

note that the transformation matrix for the composition is

$$T(\vec x)=\begin{pmatrix} \cos \alpha &- \sin \alpha&0\\ \sin \alpha & \cos \alpha&0 \end{pmatrix}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} x \cos \alpha - y \sin \alpha\\ x \sin \alpha + y \cos \alpha \end{pmatrix}$$