Question

The foot of the perpendicular drawn from a point with position vector $\hat{i}+4\hat{k}$ on the line joining the points $\hat{j}+3\hat{k}$, $2\hat{i}-3\hat{j}+\hat{k}$ is

• A. $4\hat{i}+5\hat{j}+5\hat{k}$
• B. $\frac{1}{3}(\hat{i}+\hat{j}+\widehat{8k})$
• C. $4\hat{i}+4\hat{j}-5\hat{k}$
• D. $4\hat{i}-5\hat{j}+5\hat{k}$

Answer 1

The correct option is B $\frac{1}{3}(\hat{i}+\hat{j}+\widehat{8k})$

$\begin{array}{c}\frac{x}{2}=\frac{y-1}{-4}=\frac{z-3}{-2}\\ \frac{x}{1}=\frac{y-1}{-2}=\frac{z-3}{-1}\\ Directionratioof\overrightarrow{AB}:r-1,-2r+1,-r+1\\ \left(r-1\right)-2\left(-2r+1\right)-\left(-r-1\right)=0\\ r=\frac{1}{3}\\ footofperpendicular\left(\frac{1}{3},\frac{1}{3},\frac{8}{3}\right)\\ Now,\\ \frac{1}{3}\left(\hat{i}+\hat{j}+8\hat{k}\right)\\ Hence,\\ optionBiscorrectanswer.\end{array}$