Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Direction ratios of the given line are $1,2,2$.

Direction cosines of the line are $l=\frac{1}{\sqrt{1^2+2^2+2^2}}=\frac{1}{3}$, $m=\frac{2}{3}$, $n=\frac{2}{3}$.

The coordinates of the given points are $P(2,-1,0)$ and $Q(3,2,-1)$.

The projection of the line segment $PQ$ on the given line is $|(x_2-x_1)l+(y_2-y_1)m+(z_2-z_1)n|$.

Substituting the values, we get:

Projection $=\left|(3-2)\left(\frac{1}{3}\right)+(2-(-1))\left(\frac{2}{3}\right)+(-1-0)\left(\frac{2}{3}\right)\right|$

$=\left|1\left(\frac{1}{3}\right)+3\left(\frac{2}{3}\right)-1\left(\frac{2}{3}\right)\right|$

$=\left|\frac{1}{3}+\frac{6}{3}-\frac{2}{3}\right|$

$=\frac{5}{3}$

Answer: $\frac{5}{3}$