The equations of the given lines are
\(\vec r = (\hat i +\hat j +\hat k) + \lambda (\hat i - \hat j + \hat k) \) and
\(\vec r = 2\hat i - \hat j - \hat k + \mu (2\hat i + \hat j + 2\hat k)\)
It is known that the shortest distance between the lines \(\vec r = \vec{a_1} + \lambda (\vec {b_1}) \) and \(\vec r=\vec{a_2} + \mu \vec {b_2}\) is given by,
Comparing the given equations, we obtain
Substituting all the values in equation (1), we obtain
Therefore, the shortest distance between the two lines is \(\frac{3\sqrt 2}2\) units.
