Chapter 8.8, Problem 3CFU

To determine

To state: whether the result obtained after multiplying a vertex matrix by transformational matrix $T$ is same as multiplying by $U$ and then by $V$ .

Answer to Problem 3CFU

Yes

Explanation of Solution

Given information:

  $\begin{array}{ccc}T=\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ -1\end{array}\end{array}\right]& U=\left[\begin{array}{ccc}\begin{array}{l}1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ -1\end{array}\end{array}\right]& V=\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 1\end{array}\end{array}\right]\end{array}$

Calculation:

Let the vertex matrix be $X$ . Multiply matrices $TX$ as shown below.

  $\begin{array}{c}TX=\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ -1\end{array}\end{array}\right]\cdot \left[\begin{array}{l}x\\ y\\ z\end{array}\right]\\ =\left[\begin{array}{c}-x\\ y\\ -z\end{array}\right]\end{array}$

Now multiply $X$ first by $U$ and then multiply the result obtained with $V$ as shown below.

  $\begin{array}{c}V\cdot \left(UX\right)=\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 1\end{array}\end{array}\right]\cdot \left(\left[\begin{array}{ccc}\begin{array}{l}1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ -1\end{array}\end{array}\right]\cdot \left[\begin{array}{l}x\\ y\\ z\end{array}\right]\right)\\ =\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 1\end{array}\end{array}\right]\cdot \left[\begin{array}{l}x\\ y\\ -z\end{array}\right]\\ =\left[\begin{array}{c}-x\\ y\\ -z\end{array}\right]\end{array}$

Thus the result obtained after multiplying a vertex matrix by transformational matrix $T$ is same as multiplying by $U$ and then by $V$ .