Skew lines

Skew lines are two lines in three-dimensional Euclidean space that are neither parallel nor intersecting.

For lines $a_1+t{m}_1$ and $a_2+u{m}_2$ with nonparallel direction vectors, their shortest separation is the absolute orthogonal projection of $a_2-a_1$ onto their common normal $m_1\times m_2$. Components in the span of $m_1,m_2$ can be cancelled by choosing $t,u$, while the normal component cannot. The same formula gives zero for intersecting nonparallel lines. Parallel lines instead have separation $|(a_2-a_1)\times m_1|/|m_1|$.

For three skew lines $L_i$ and a point $\mathbf{y}$ outside them, let ${\Pi }_i$ be the plane through $\mathbf{y}$ and $L_i$. A line through $\mathbf{y}$ meeting all three $L_i$ must lie in all three ${\Pi }_i$. Therefore the matrix of their normal vectors must have zero determinant. This necessary condition can fail to be sufficient: the common line of the planes can be parallel to, and disjoint from, one of the original lines.

For example, take $L_1=(s,0,1)$, $L_2=(1,s,0)$, $L_3=(0,1,s)$ and $\mathbf{y}=(2,1,0)$. The corresponding planes have equations $r_2+r_3=1$, $r_3=0$, $r_2=1$. Their common line is $(s,1,0)$, which never meets $L_1$. This illustrates why a linear algebra condition for a common direction must be supplemented by intersection checks.

For lines ${\mathbf{a}}_1+\lambda {\mathbf{u}}_1$ and ${\mathbf{a}}_2+\mu {\mathbf{u}}_2$ with ${\mathbf{u}}_1\times {\mathbf{u}}_2\ne 0$, their shortest distance is

For nonparallel directions ${\mathbf{b}}_1,{\mathbf{b}}_2$ and points ${\mathbf{a}}_1,{\mathbf{a}}_2$, let $s_1,s_2$ be the independent line parameters at the closest points. Put $\Delta ={\mathbf{a}}_2-{\mathbf{a}}_1$ and $D=|{\mathbf{b}}_1\times {\mathbf{b}}_2{|}^2>0$. Perpendicularity of the connecting vector to both directions gives a two-by-two linear system. Solving it and subtracting yieldsParticles with positions ${\mathbf{a}}_j+{\mathbf{b}}_j(t-t_j)$ reach these two points simultaneously exactly when $t_1-t_2=s_2-s_1$. Otherwise, their closest approach of two uniformly moving particles can exceed the geometric line distance. The formula requires nonparallel directions; it is not an identity obeyed by arbitrary fixed starting times.