Question

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

Answer 1

$\begin{array}{c}\Rightarrow L_1=\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z+1}{1}\\ \Rightarrow {\pi }_1=2x-y+z=1\\ \Rightarrow \sin \theta =\frac{2+1+1}{\sqrt{6\sqrt{3}}}\\ \Rightarrow \frac{4}{3\sqrt{2}}=\frac{2\sqrt{2}}{3}\end{array}$

Angle between line and plane is $\theta ={\sin }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$