Condition for Perpendicular Lines

For $y=m_1x+c_1$ and $y=m_2x+c_2$ to be perpendicular, the angle between them must be $\pi/2$: taking the cotangent of $\tan\varphi=\dfrac{m_2-m_1}{1+m_1m_2}$ gives $\cot\varphi=\dfrac{1+m_1m_2}{m_2-m_1}$, and $\cot(\pi/2)=0$ forces $1+m_1m_2=0$, i.e.

In general form, $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ are perpendicular exactly when $a_1a_2+b_1b_2=0$.

Two ready-made results, mirroring the parallel case:

1. Every line perpendicular to $ax+by+c=0$ has the form $bx-ay=k$.
2. The line through $(x_1,y_1)$ perpendicular to $ax+by+c=0$ is $bx-ay=bx_1-ay_1$.

Summary table: