Question

Find the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$.

• A. $\frac{1}{\sqrt{6}}$
• B. $\frac{1}{\sqrt{20}}$
• C. $\frac{1}{\sqrt{32}}$
• D. None of these

Answer 1

The correct option is A $\frac{1}{\sqrt{6}}$

$n_1\times n_2=(2i+3j+4k)\times (3i+4j+5k)$
$=-i+2j-k$

And

$\overrightarrow{a_1}-\overrightarrow{a_2}$
$=(2-1)i+(4-2)j+(5-3)k$
$=i+2j+2k$

$d=\frac{(\overrightarrow{a_1}-\overrightarrow{a_2}).(n_1\times n_2)}{|n_1\times n_2|}$

$=\frac{(1+2j+2k).(-i+2j-k)}{\sqrt{6}}$

$=\frac{-1+4-2}{\sqrt{6}}$

$=\frac{1}{\sqrt{6}}$.