Question

The distance between a point $P$ whose position vector is $5\hat{i}+\hat{j}+3\hat{k}$ and the line $\overrightarrow{r}=(3\hat{i}+7\hat{j}+\hat{k})+\lambda (\hat{j}+\hat{k})$ is

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

Answer: (D)

$6$

$\overrightarrow{r}=3\overrightarrow{i}+(7+\lambda )\hat{j}+(1+\lambda )\hat{k}$
$(\hat{r}-\hat{a})=-2\hat{i}+(6+\lambda )\hat{j}+(\lambda -2)\hat{k}$
$(\overrightarrow{r}-\overrightarrow{a})\perp (\hat{j}+\hat{k})$
$(\overrightarrow{r}-\overrightarrow{a}).(\hat{j}+\hat{k})=0$
$6+\lambda +\lambda -2=0$
$\lambda =-2$
$(\overrightarrow{r}-\overrightarrow{a})=-2\hat{i}+4\hat{j}-4\hat{k}$
$|\overrightarrow{r}-\overrightarrow{a}|=6$