Question

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

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is D $6$

The ratio of $\overrightarrow{PQ}=\left(-2,6+\lambda ,\lambda -2\right)$

$\overrightarrow{PQ}\perp line$

$\therefore -2\times 0+\left(6+\lambda \right)\times 1+\left(\lambda -2\right)\times 1=0$

$6+\lambda +\lambda -2=0$

$2\lambda =-4$

$\therefore \lambda =-2$

Now,

The coordinates of points $P(5,1,3)$ and $Q(3,5,-1)$

Now,

Distance of points $PQ=\sqrt{{\left(2\right)}^2+{\left(-4\right)}^2+{\left(4\right)}^2}$

$=\sqrt{36}=6$

Hence, the correct answer is $D$