User:KohanX/Formulae

The distance between two lines $\mathbf {A}$ and $\mathbf {B}$, where $\mathbf {a}$ and $\mathbf {b}$ are the respective directions, and $\mathbf {a^{0}}$ and $\mathbf {b^{0}}$ are an arbitrary point along said lines, respectively. Both the directions and points are vectors, of course.

$d(\mathbf {A} ,\mathbf {B} )={\bigg |}(\mathbf {a^{0}} -\mathbf {b^{0}} )+{\frac {((\mathbf {a} \cdot \mathbf {b} )(\mathbf {b} \cdot (\mathbf {a^{0}} -\mathbf {b^{0}} ))-(\mathbf {a} \cdot (\mathbf {a^{0}} -\mathbf {b^{0}} )))\mathbf {a} -((\mathbf {b} \cdot (\mathbf {a^{0}} -\mathbf {b^{0}} ))-(\mathbf {a} \cdot \mathbf {b} )(\mathbf {a} \cdot (\mathbf {a^{0}} -\mathbf {b^{0}} )))\mathbf {b} }{1-(\mathbf {a} \cdot \mathbf {b} )^{2}}}{\bigg |}$