Question

The measure of the angle between the line $\overrightarrow{r}=(2,-3,1)+k(2,2,1);k\in R$ and the plane $2x-2y+z+7=0$ is

• A. ${\cos }^{-1}\frac{1}{9}$
• B. ${\tan }^{-1}\frac{1}{4\sqrt{5}}$
• C. ${\sin }^{-1}\frac{1}{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is B ${\tan }^{-1}\frac{1}{4\sqrt{5}}$

Direction ratios of the given line are $(2,2,1)$
Direction ratios of the normal of given plane are $(2,-2,1)$
Hence, the required angle is
$\sin \theta =\frac{2\times 2+2\times (-2)+1\times 1}{\sqrt{4+4+1}\sqrt{4+4+1}}$
$\Rightarrow \sin \theta =\frac{1}{9}$
$\tan \theta =\frac{1}{\sqrt{80}}$
$\therefore \theta ={\tan }^{-1}\frac{1}{4\sqrt{5}}$