Question

If $P(3,4,5),Q(4,6,3),R(-1,2,4),S(1,0,5)$then the projection of $RS$ on $PQ$ is

• A. $-\frac{2}{3}$
• B. $-\frac{4}{3}$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

The correct option is B

$-\frac{4}{3}$

Explanation for the correct option.

Given, $P(3,4,5),Q(4,6,3),R(-1,2,4),S(1,0,5)$be any four points.
Then direction ratios of the line $PQ$ are $(1,2,-2)$.
The direction ratios of the line $RS$ are $(2,-2,1)$.

We know that projection of $\overrightarrow{a}$ on $\overrightarrow{b}$is given by $\frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{b}\right|}$.
$\begin{array}{rcl}\mathrm{The}\mathrm{projection}\mathrm{of}\mathrm{RS}\mathrm{on}\mathrm{PQ}& =& \frac{(\mathrm{i}+2\mathrm{j}-2\mathrm{k})·(2\mathrm{i}-2\mathrm{j}+\mathrm{k})}{3}\\ & =& \frac{2-4-2}{3}\\ & =& -\frac{4}{3}\end{array}$

Hence, option B is correct.