Question

The length of the projection of the line segment joining the points $(5,-1,4)$ and $(4,-1,3)$ on the plane, $x+y+z=7$ is

• A. $\frac{2}{\sqrt{3}}$
• B. $\frac{2}{3}$
• C. $\frac{1}{3}$
• D. $\sqrt{\frac{2}{3}}$

Answer 1

The correct option is D $\sqrt{\frac{2}{3}}$


Dr’s of $AB=(1,0,1)$
and dr’s of normal to plane $=(1,1,1)$
$\Rightarrow \cos (\frac{\pi }{2}-\theta )=\frac{1+1}{\sqrt{2}\cdot \sqrt{3}}$
$\Rightarrow \sin \theta =\sqrt{\frac{2}{3}}$
$AB=\sqrt{2}$
Length of projection $=AB\cos \theta =\sqrt{2}\cdot \sqrt{1-\frac{2}{3}}$
$\therefore$ Length of projection $=\sqrt{\frac{2}{3}}$