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

Two (affine) lines ${\displaystyle {}G,H\subseteq \mathbb {R} ^{3}}$ 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 ${\displaystyle {}u}$ perpendicular to this plane. We can compute one such vector, the normal vector, with the cross product. Let

${\displaystyle {}G=P+\mathbb {R} v\,,}$

and

${\displaystyle {}H=Q+\mathbb {R} w\,.}$

The linear system of equations

${\displaystyle {}P-Q=av+bw+cu\,}$

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