= 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 yields
$$
s_2-s_1=-\frac{\Delta\cdot[(\mathbf b_1\times\mathbf b_2)\times(\mathbf b_1-\mathbf b_2)]}{D}.
$$
Particles 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.
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