Skew lines/R^3/Distance/Linear system/Example

Two (affine) lines $G,H\subseteq {ℝ}^3$ are called skew if they do not have any point in common and if they are also not parallel, meaning their vectors are linearly independent. Then, these vectors generate a plane; there exists a vector $u$ perpendicular to this plane. We can compute one such vector, the normal vector, with the cross product. Let

$G=P+ℝv,$

and

$H=Q+ℝw.$

The linear system of equations

$P-Q=av+bw+cu$

has a unique solution $(a,b,c)\in {ℝ}^3$. Here, $P-av\in G$ and $Q+bw\in H$ are the dropped perpendicular feet, where the distance of the lines is obtained, according to fact. This distance is $\Vert cu\Vert$.