Question

The distance of the point $B$ with position vector $i+2j+3k$ from the line passing through the point $A$ with position vector $4i+2j+2k$ and parallel to the vector $2i+3j+6k$ is

• A. $\sqrt{10}$
• B. $\sqrt{5}$
• C. $\sqrt{6}$
• D. none of these

Answer 1

The correct option is A $\sqrt{10}$

Equation of the line passing through the point $A(4i+2j+2k)$ and parallel to $(2i+3j+6k)$ is
$\overrightarrow{r}=(4i+2j+2k)+t(2i+3j+6k)$
any point on the line is of the form $(4+2t,2+3t,2+6t)$
let $BC$ is perpendicular to the given line, $B(i+2j+3k)$ and $C(4+2t,2+3t,2+6t)$
applying perpendicularity condition
$(2t+3)2+\left(3t\right)3+(6t-1)6=0$
$\Rightarrow t=0$
$\therefore$ $BA$ is perpendicular to the line
Hence,required distance is $\sqrt{10}$