Question

Find the measure of the angle between the line $\frac{x-2}{2}=\frac{y-2}{-3}=\frac{z-1}{2}$ and the plane $2x+y-3z+4=0$.

Answer 1

line has direction ratios $\overrightarrow{l}=(2,-3,2)$ and normal to plane has d.r.s $\overrightarrow{r}=(2,1,-3)$ Angle between line and normal to plane,

$\theta ={\cos }^{-1}\left(\frac{\overrightarrow{l}.\overrightarrow{r}}{|\overrightarrow{l}||\overrightarrow{r}|}\right)\Rightarrow \theta ={\cos }^{-1}\left(\frac{4-3-6}{\sqrt{17}\sqrt{14}}\right)={\cos }^{-1}\left(\frac{5}{\sqrt{17\times 14}}\right)$

line between line and plane would be

complimentary to this $=90-\theta =90-{\cos }^{-1}\left(\frac{-5}{\sqrt{238}}\right)$