Question

The length of projection of the line segment joining the point $(1,0,-1)$ and $(-1,2,2)$ to the plane $x+3y-5z=6$, is equal to -

• A. 2
• B. $\sqrt{\frac{271}{53}}$
• C. $\sqrt{\frac{472}{31}}$
• D. $\sqrt{\frac{474}{35}}$

Answer 1

The correct option is D $\sqrt{\frac{474}{35}}$

$\overrightarrow{PQ}=-2\hat{i}+2\hat{j}+3\hat{k}$

$|\overrightarrow{PQ}|=\sqrt{4+4+9}=\sqrt{17}$



Angle between the line and plane

$\sin \theta =\left|\frac{-2.1+2.3-5.3}{\sqrt{1+9+25}\sqrt{17}}\right|=\frac{11}{\sqrt{595}}$

$\cos \theta =\frac{\sqrt{474}}{\sqrt{595}}$,
Projection $=|\overrightarrow{PQ}|\cos \theta =\sqrt{\frac{474}{35}}$