Question

A line L${}_1$ passing through a point with position vector $\overrightarrow{p}=\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k}$ and parallel to $\overrightarrow{a}=\overrightarrow{i}+2\overrightarrow{j}+3\overrightarrow{k}$, Another line $L$${}_2$ passing through a point with position vector $\overrightarrow{q}=2\overrightarrow{i}+3\overrightarrow{j}+\overrightarrow{k}$ and parallel to $\overrightarrow{b}=3\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k}$.

The minimum distance of origin from the plane passing through the point with position vector $\overrightarrow{p}$ and perpendicular to the line $L$ ${}_2$ is

• A. $\sqrt{14}$
• B. $\frac{7}{\sqrt{14}}$
• C. $\frac{11}{\sqrt{14}}$
• D. None of these

Answer 1

Answer: (C)

$\frac{11}{\sqrt{14}}$

Equation of plane passing through $\overrightarrow{p}$ and perpendicular to line $L_2$ is given by,
$(\overrightarrow{r}-\overrightarrow{p})\cdot \overrightarrow{b}=0$
$\Rightarrow (x-1)3+(y-2)1+(z-3)2=0$
$\Rightarrow 3x+y+2z-11=0$
Hence, minimum (shortest) distance of this plane from origin is,
$=\frac{11}{\sqrt{3^2+1^2+2^2}}=\frac{11}{\sqrt{14}}$