15-00-ExampleOfRotationMatrix.tex

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\pmtitle{example of rotation matrix}
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\begin{document}
You can use rotation matrices to show that if the slope of one line is $m$, then the slope of the line perpendicular to it is $\frac{-1}{m}$:

Let $L$ be a line with a slope of $m$ passing through the origin. The rotation matrix $R_{\frac{\pi}{2}}$ rotates $L$ into a line $L^\prime$ perpendicular to $L$:

$$ R_{\pi/2} = 
\begin{pmatrix}
0 & -1 \\
1 & 0 \\
\end{pmatrix}
$$

Every point on $L$ can be represented as a multiple of the point 
$ \vec{p} = \begin{pmatrix} 1 \\ m \end{pmatrix} $.

Notice $ \vec{p}^\prime = R_{\frac{\pi}{2}} \vec{p} = \begin{pmatrix} -m \\ 1 \end{pmatrix} $. Since every point on $L^\prime$ can be represented as a multiple of the point $\vec{p}^\prime$, the slope of $L^\prime$ is $\frac{-1}{m}$.
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