Question

Find the angle between lines $x = 2$ and $x - 3y = 6$.

Answer 1

$\arctan(3)$

Answer 2

The line $x=2$ is a vertical line, so its angle of inclination is $90^\circ$. The line $x-3y=6$ can be rewritten in slope-intercept form as $y = \frac{1}{3}x - 2$. Thus, its slope is $m = \frac{1}{3}$. The angle $\theta$ between a vertical line and a line with slope $m$ is given by $\tan\theta = \left|\frac{1}{m}\right|$. Substituting $m=\frac{1}{3}$, we get $\tan\theta = \left|\frac{1}{1/3}\right| = 3$. Therefore, the angle between the lines is $\theta = \arctan(3)$.