Petrovcic2x

2022-06-24

Let ${T}_{1}$ be a reflection of ${\mathbb{R}}^{3}$ in the xy plane, ${T}_{2}$ is a reflection of ${\mathbb{R}}^{3}$ in the xz plane. What is the standard matrix of transformation ${T}_{2}{T}_{1}$?
Here's my thinking so far:
Since the standard matrix for reflections in xy is
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$
Similarly, standard matrix for orthogonal projection in the xz plane is
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]$
I could multiply
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\ast \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$
to yield
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
Could someone confirm for me if this is a valid approach?

Braylon Perez

Reflection in the xy plane has matrix
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$
and similar for reflection in the xz plane.
Your matrices, as you say once, are for projection, rather than reflection. If you reflect in the xy plane, the x and y values stay the same, but the z-value becomes its negative.
Your idea of multplying the matrices is correct.

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