Find the shortest distance between the lines

$\overrightarrow{r}$ = $\hat{i} + \hat{j} + \lambda(2\hat{i} - \hat{j} + \hat{k}) $

$\overrightarrow{r}$ = $2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k}) $

The correct answer is $\dfrac{10}{\sqrt{59}}$

Here,

$\overrightarrow{a_{1}}$ = $\hat{i} + \hat{j}$, $\overrightarrow{b_{1}}$ = $2\hat{i} - \hat{j} + \hat{k} $

$\overrightarrow{a_{2}}$ = $2\hat{i} + \hat{j} - \hat{k}$, $\overrightarrow{b_{2}}$ = $3\hat{i} - 5\hat{j} + 2\hat{k} $

∴ $\overrightarrow{a_{2}}$ - $\overrightarrow{a_{1}}$ = $\hat{i} - \hat{k}$



$\overrightarrow{b_{1}}$ x $\overrightarrow{b_{2}}$ =
$\left| \begin{matrix}

\hat{i} & \phantom{-}\hat{j} & \phantom{-}\hat{k} \\

2 & \phantom{-}-1 & \phantom{-}1 \\

3 & \phantom{-}-5 & \phantom{-}2 \\

\end{matrix} \right|$ = $3\hat{i} - \hat{j} -7\hat{k}$


∴ $|\overrightarrow{b_{1}}$ x $\overrightarrow{b_{2}} |$ = $\sqrt{9 + 1 + 49}$ = $\sqrt{59}$


∴ The shortest distance between the given lines is



$d$ = $| \dfrac{(\overrightarrow{b_{1}} \text{x} \overrightarrow{b_{2}}). (\overrightarrow{a_{2}} - \overrightarrow{a_{1}})}{|\overrightarrow{b_{1}} \text{x} \overrightarrow{b_{2}}|}|$

= $\dfrac{| 3 - 0 + 7|}{\sqrt{59}}$

= $\dfrac{10}{\sqrt{59}}$