If we have two skew lines in $\mathbb R^3$, $\vec r_{1} = \vec a + \lambda\vec d_1$ and $\vec r_{2} = \vec b + \mu\vec d_2$ then at their closest point, the difference vector $\vec r_2 - \vec r_1$ is perpendicular to both $\vec r_1$ and $\vec r_2$, so:
$(\vec r_2 - \vec r_1) \cdot \vec d_1 = 0$
$(\vec r_2 - \vec r_1) \cdot \vec d_2 = 0$
from which we have two eqns in $\lambda$ and $\mu$ which we can solve to find the required minimum $\vec r_2 - \vec r_1$.
Is this reasoning correct ?