 Synchronizing the closest points of two lines

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.