Question

The distance between the line

$\overset{\rightarrow}{r}$ = (2$\widehat{i}$ – 2$\widehat{j}$ + 3$\widehat{k}$) +l($\widehat{i}$–$\widehat{j}$+ 4$\widehat{k}$) & the plane

$\overset{\rightarrow}{r}$.($\widehat{i}$+ 5$\widehat{j}$ + $\widehat{k}$) = 5 is :

• A. $\frac{10}{3\sqrt{3}}$
• B. $\frac{10}{3}$
• C. $\frac{10}{9}$
• D. None of these

Answer 1

Answer: (A)

$\frac{10}{3\sqrt{3}}$

Answer 2

Given line is

Ž $\overset{\rightarrow}{r}$= (2$\widehat{i}$ – 2$\widehat{j}$ + 3$\widehat{k}$) + l($\widehat{i}$ –$\widehat{j}$ + 4$\widehat{k}$) .....(i)

& $\overset{\rightarrow}{r}$.($\widehat{i}$ +5$\widehat{j}$ + $\widehat{k}$) = 5 ....(ii)

By (i) Ž $\left\{ \begin{aligned} & \overset{\rightarrow}{a} = (2\widehat{i}–2\widehat{j} + 3\widehat{k}) \\ & \overset{\rightarrow}{b} = (\widehat{i}–\widehat{j} + 4\widehat{k}) \end{aligned} \right.\$

By (ii) Ž $\overset{\rightarrow}{n}$ = ($\widehat{i}$ +5$\widehat{j}$ + $\widehat{k}$)

Q $\overset{\rightarrow}{b}$.$\overset{\rightarrow}{n}$ = 0

Therefore, the line is parallel to the plane. Thus, the distance between the line & the plane is equal to the length of the perpendicular from a point $\overset{\rightarrow}{a}$ = (2$\widehat{i}$ – 2$\widehat{j}$ + 3$\widehat{k}$) on the line to the given plane.

Hence, the required distance

=$\left| \frac{(2\widehat{i}–2\widehat{j} + 3\widehat{k}).(\widehat{i} + 5\widehat{j} + \widehat{k})–5}{\sqrt{1 + 5^{2} + 1}} \right|$

= $\frac{10}{3\sqrt{3}}$