# Slopes and lines

Given two lines having slope $m_1$ and $m_2$, the angle between them is given by $\tan(\theta)=\frac{m_2- m_1}{1+m_1m_2}$

Does the order of $m_1$ and $m_2$ matter here, and if so what is the significance? Graphical aid would be helpful.

• What do you mean with order? – Fimpellizieri Nov 4 '16 at 20:54
• Like would changing $m_2$-$m_1$ to $m_1$-$m_2$ change anything? I know it does....but I'm not sure what exactly.... – SaitamaSensei Nov 4 '16 at 20:56
• It does: it changes the sign of $\tan(\theta)$; you can check that directly in the formula. This means there is an implicit positive orientation (counterclockwise) and a negative orientation (clockwise). When you reverse the order, you reverse orientation. – Fimpellizieri Nov 4 '16 at 21:02

Let's see what happens if we switch $m_1$ and $m_2$. First call $\theta_1$ an angle such that $\tan\theta_1 = \frac{m_2-m_1}{1+m_1m_2}$ and $\theta_2$ such that $\tan\theta_2 = \frac{m_1-m_2}{1+m_2m_1}$. Immediately you can see that $\tan\theta_1 = -\tan\theta_2$ and by tangens being odd function, we get $\tan\theta_1 = \tan(-\theta_2)$. If we restrict our $\theta$'s to $\langle-\pi/2,\pi/2\rangle$ (which we usually do in this case since we want the acute angle between lines) we get that $\theta_2 = -\theta_1$ so it is "the same" angle but measured in opposite directions (clockwise or counterclockwise).
In conclusion, $\theta_1$ is the angle between lines $l_1$ and $l_2$ measured from line $l_1$ to $l_2$ and $\theta_2$ is the angle between $l_1$ and $l_2$ measured from $l_2$ to $l_1$. They are of the same absolute value, but opposite orientations.
If you don't want to be bothered with orientation, you can change formula to $\tan\theta =\left|\frac{m_2-m_1}{1+m_1m_2}\right|$ and you will always get a positive angle and switching $m_1$ and $m_2$ won't change it since $|m_2-m_1|=|m_1-m_2|$.
Note that $m_1=\tan \theta_1$ and $m_1=\tan \theta_2$ are the tangents of the angles $\theta_1$ and $\theta_2$ between the lines and the $x$ axis. So: $$\frac{m_2-m_1}{1+m_1m_2}=\frac{\tan \theta_2-\tan \theta_1}{1+\tan \theta_1\tan \theta_2}=\tan (\theta_2-\theta_1)$$ and: $$\frac{m_1-m_2}{1+m_2m_1}=\frac{\tan \theta_1-\tan \theta_2}{1+\tan \theta_2\tan \theta_1}=\tan (\theta_1-\theta_2)$$ so the two formulas give two possible orientation for the angle $\theta$ between two lines.