Chapter 2.4, Problem 30E

To determine

To show: dilation with scale factor of $-1$ and that of $Ro{t}_{180}$ has same results.

Answer to Problem 30E

See explanation.

Explanation of Solution

Given information:

Scale factor = $-1$

  $R_{180}=\left[\begin{array}{cc}\begin{array}{l}-1\\ 0\end{array}& \begin{array}{l}0\\ -1\end{array}\end{array}\right]$

Calculation:

Let the position matrix $\left(P\right)$ be $\left[\begin{array}{cc}\begin{array}{l}{x}_1\\ y_1\end{array}& \begin{array}{l}{x}_2\\ y_2\end{array}\end{array}\right]$ .

  $\begin{array}{c}P\circ R_{180}=\left[\begin{array}{cc}\begin{array}{l}{x}_1\\ y_1\end{array}& \begin{array}{l}{x}_2\\ y_2\end{array}\end{array}\right]\left[\begin{array}{cc}\begin{array}{l}-1\\ 0\end{array}& \begin{array}{l}0\\ -1\end{array}\end{array}\right]\\ =\left[\begin{array}{cc}\begin{array}{l}-x_1\\ -y_1\end{array}& \begin{array}{l}-x_2\\ -y_2\end{array}\end{array}\right]\\ =-1\left[\begin{array}{cc}\begin{array}{l}{x}_1\\ y_1\end{array}& \begin{array}{l}{x}_2\\ y_2\end{array}\end{array}\right]\\ =\text{Dilation with scale factor of }-1\end{array}$