Gradient intersecting of lines other than right angle>.

Question

Given two straight lines:

1. $$y=mx+c$$
2. $$y=m_1x+c$$

When they intersect at right angle, their relation is that $mm_1=-1.$

What is their relationship (concerning gradient of lines) if they intersect other than at right angle?

Answer 1

In general, we have that \begin{align} mm'+1 = \sqrt{m^2+1}\sqrt{(m')^2+1}\cos\theta \end{align} where $\theta$ is the angle between the two intersecting lines.

In the special case where the intersection is a right angle, then we have that \begin{align} mm'+1 = 0 \end{align} which is what you have.

Answer 2

Well it follows from $\tan$ of difference of angles formula. Let $\tan(\theta) = m_1$ and $\tan(\phi) = m_2$. Then $\delta =|\theta - \phi|$ is one of the angle between the lines (other is $\pi - |\theta - \phi|$)

$$\tan(\delta)=\tan(|\theta - \phi|) = \left|\frac{\tan\theta - \tan \phi}{1-\tan\theta \tan \phi}\right|=\left|\frac{m_1-m_2}{1+m_1m_2}\right|$$