Let $T_1$ be a reflection of $\Bbb{R}^3$ in the xy plane, $T_2$ is a reflection of $\Bbb{R}^3$ in the xz plane. What is the standard matrix of transformation $T_2T_1$?

Here's my thinking so far:

Since the standard matrix for reflections in xy is $$\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}$$

Similarly, standard matrix for orthogonal projection in the xz plane is $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&1\\\end{bmatrix}$$

I could multiply $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&1\\\end{bmatrix}*\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}$$

to yield $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\\\end{bmatrix}$$

Could someone confirm for me if this is a valid approach? Cheers.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.