Question

Foot of $\perp$ of the point $P(1,2,3)$ on the line $\overline{r}=\overline{a}+\lambda \overline{b}=(6\hat{i}+7\hat{j}+7\hat{k})+\lambda (3\hat{i}+2\hat{j}-2\hat{k})$ is

• A. $3\hat{i}-5\hat{j}+9\hat{k}$
• B. $3\hat{i}+5\hat{j}-9\hat{k}$
• C. $3\hat{i}+5\hat{j}+9\hat{k}$
• D. None of these

Answer 1

Answer: (C)

$3\hat{i}+5\hat{j}+9\hat{k}$

Using fact Foot of $\perp$ of a point P(a) in the
$\overline{r}=\overline{a}+\lambda \overline{b}$ is given by
$\overline{a}-\overline{a}-\left(\frac{a-\overline{a}}{{\overline{b}}^2}\right)\overline{b}$
$=(6\hat{i}+7\hat{j}+7\hat{k})$
$=\left(\frac{\left[\right(6\hat{i}+7\hat{j}+7\hat{k})-(\hat{i}+2\hat{j}+3\hat{k}\left)\right][3\hat{i}+2\hat{j}-2\hat{k}]}{3^2+2^2+(-2{)}^2}\right)$
$=(6\hat{i}+7\hat{j}+7\hat{k})$
$\left(\frac{\left[\right(5+\hat{i}+5\hat{j}+4\hat{k}\left)\right(3\hat{i}+2\hat{j}-2\hat{k}\left)\right][3\hat{i}+2\hat{j}-2\hat{k}]}{17}\right)$
$=(6\hat{i}+7\hat{j}+7\hat{k})-\frac{(15+10-8)}{17}(3\hat{i}+2\hat{j}-2\hat{k})$
$(6\hat{i}+7\hat{j}+7\hat{k})-(3\hat{i}+2\hat{j}-2\hat{k})$
$=3\hat{i}+5\hat{j}+9\hat{k}$
$\therefore$ required foot of $\perp$ of the point $\hat{i}+2\hat{j}+3\hat{3}$ on $\overline{r}=(6\hat{i}+7\hat{j}+7\hat{k})+\lambda \overline{b}$ is given by the PV $3\hat{i}+5\hat{j}+9\hat{k}$.