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 $\frac{-1}{m}$