The shortest distance between the lines (x + 2)/(1) = (y)/(- 2) =...

Given,

$\dfrac{x + 2}{1} = \dfrac{y}{- 2} = \dfrac{z - 5}{2}$

$\dfrac{x - 4}{1} = \dfrac{y - 1}{2} = \dfrac{z + 3}{0}$

Now we know that,

Shortest distance between two skew lines is given by,

$SD = \left|\dfrac{\left(\overset{\to}{{a}_{2}} - \overset{\to}{{a}_{1}}\right)\left(\overset{\to}{{n}_{1}} \times \overset{\to}{{n}_{2}}\right)}{\left|\overset{\to}{{n}_{1}} \times \overset{\to}{{n}_{2}}\right|}\right|$

Now finding, \overset{\to}{{n}_{1}} \times \overset{\to}{{n}_{2}} = \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 1 & - 2 & 2 \\ 1 & 2 & 0\end{cases}\right|

⇒ \overset{\to}{{n}_{1}} \times \overset{\to}{{n}_{2}} = - 4\overset{^}{i} + 2\overset{^}{j} + 4\overset{^}{k}

And \overset{\to}{{a}_{2}} - \overset{\to}{{a}_{1}} = \left(4\overset{^}{i} + \overset{^}{j} - 3\overset{^}{k} - \left(- 2\overset{^}{i} + 0\overset{^}{j} + 5\overset{^}{k}\right)\right) = 6\overset{^}{i} + \overset{^}{j} - 8\overset{^}{k}

Now putting the value in formula we get,

SD = \left|\dfrac{\left(6\overset{^}{i} + \overset{^}{j} - 8\overset{^}{k}\right)\left(- 4\overset{^}{i} + 2\overset{^}{j} + 4\overset{^}{k}\right)}{\sqrt{16 + 16 + 4}}\right|

$⇒ SD = \left|\dfrac{- 24 + 2 - 32}{6}\right| = \dfrac{54}{6} = 9$