example of rotation matrix

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 $L$:

Every point on $L$ can be represented as a multiple of the point$\overrightarrow{p}=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill m\hfill \end{array}\right)$.

Notice ${\overrightarrow{p}}^{\prime }=R_{\frac{\pi }{2}}\overrightarrow{p}=\left(\begin{array}{c}\hfill -m\hfill \\ \hfill 1\hfill \end{array}\right)$. Since every point on $L^{\prime }$ can be represented as a multiple of the point ${\overrightarrow{p}}^{\prime }$, the slope of $L^{\prime }$ is $\frac{-1}{m}$.