Question

If $P=(3,4,5),Q=(4,6,3),R=(-1,2,4)$ and $S=(1,0,5)$ are four points then the projection of segment $RS$ on line $PQ$ is

• A. $\frac{8}{3}$
• B. $\frac{4}{3}$
• C. $4$
• D. $0$

Answer 1

The correct option is B $\frac{4}{3}$

Projection of $RS$ on $PQ=\overrightarrow{RS}\cdot \widehat{PQ}$
$=|l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1)|$
$\text{where}(l,m,n)\text{are d.c's of}PQ$
$\Rightarrow (l,m,n)=(\frac{4-3}{\sqrt{1+4+4}},\frac{6-4}{\sqrt{1+4+4}},\frac{3-5}{\sqrt{1+4+4}})\Rightarrow (l,m,n)=(\frac{1}{3},\frac{2}{3},-\frac{2}{3})$ and
$\overrightarrow{RS}=2\hat{i}-2\hat{j}+\hat{k}$
So, projection $=|\frac{2}{3}-\frac{4}{3}-\frac{2}{3}|=\frac{4}{3}\text{unit}$