Question

The projection of the line segment joining $(2,0,-3)$ and $(5,-1,2)$ on a straight line whose direction ratios are $2,4,4$, is

• A. $\frac{11}{6}$
• B. $\frac{10}{3}$
• C. $\frac{13}{3}$
• D. $\frac{13}{6}$
• E. $\frac{11}{3}$

Answer 1

The correct option is E $\frac{11}{3}$

Let $a$ be the vector joining $(2,0,-3)$ and $(5,-1,2)$
$a=(5-2)\hat{i}+(-1-0)\hat{j}+(2+3)\hat{k}$
$=3\hat{i}-\hat{j}+5\hat{k}$
and $b=2\hat{i}+4\hat{j}+4\hat{k}$
Therefore, projection $=\frac{a\cdot b}{\left|b\right|}$
$=\frac{(3\hat{i}-\hat{j}+5\hat{k})\cdot (2\hat{i}+4\hat{j}+4\hat{k})}{|2\hat{i}+4\hat{j}+4\hat{k}|}$
$=\frac{6-4+20}{\sqrt{4+16+16}}$
$=\frac{22}{6}=\frac{11}{3}$