Chapter 8.8, Problem 21E

To determine

To find: the transformational matrix for the given situation.

Answer to Problem 21E

  $\left[\begin{array}{ccc}\begin{array}{l}-0.5\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0.5\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0.5\end{array}\end{array}\right]$

Explanation of Solution

Given information:

First the transformational matrix should reflect a rectangular prism over the $yz$ -plane and then reduce its dimension by half.

Calculation:

The matrix $\left(R_{yz-plane}\right)$ that would reflect the prism over $yz$ plane and matrix $\left(H\right)$ that would reduce its dimensions by half are shown below.

  $R_{yz-plane}=\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 1\end{array}\end{array}\right]\text{ and }H=\left[\begin{array}{ccc}\begin{array}{l}0.5\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0.5\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0.5\end{array}\end{array}\right]$

The transformational matrix for the given situation will be evaluated as follows:

  $\begin{array}{l}T=\left(H\right)\left(R_{yz-plane}\right)\\ =\left[\begin{array}{ccc}\begin{array}{l}0.5\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0.5\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0.5\end{array}\end{array}\right]\left[\begin{array}{ccc}\begin{array}{l}-1\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 1\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 1\end{array}\end{array}\right]\\ =\left[\begin{array}{ccc}\begin{array}{l}-0.5\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0.5\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0.5\end{array}\end{array}\right]\end{array}$

Thus the transformational matrix is $\left[\begin{array}{ccc}\begin{array}{l}-0.5\\ 0\\ 0\end{array}& \begin{array}{l}0\\ 0.5\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 0.5\end{array}\end{array}\right]$ .