Question

Angle between line
$\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-8}{2}$ and plane $2x+y+2z+5=0$ is

• A. ${\cos }^{-1}\left(\frac{8}{9}\right)$
• B. ${\sin }^{-1}\left(\frac{8}{9}\right)$
• C. ${\cos }^{-1}\left(\frac{1}{9}\right)$
• D. ${\sin }^{-1}\left(\frac{1}{9}\right)$

Answer 1

The correct option is B ${\sin }^{-1}\left(\frac{8}{9}\right)$

Direction ratios of line and the normal of the plane are $(1,2,2)$ and $(2,1,2)$ respectively.
Angle between line and normal of the plane is
$\cos \left(\theta \right)=\left(\frac{1\times 2+2\times 1+2\times 2}{\left(\sqrt{1^2+2^2+2^2}\right)\times \left(\sqrt{2^2+1^2+2^2}\right)}\right|\cos \left(\theta \right)=\frac{8}{9}\theta ={\cos }^{-1}\left(\frac{8}{9}\right)$
Angle between line and normal of the plane is ${\cos }^{-1}\left(\frac{8}{9}\right)$.
So, angle between the line and the plane is $\frac{\pi }{2}-{\cos }^{-1}\left(\frac{8}{9}\right)={\sin }^{-1}\left(\frac{8}{9}\right)$.