Relative motion

The position and velocity of one body relative to another are the differences of their corresponding vectors in the same reference frame. Constant relative velocity gives an affine function of time, even when neither body is at rest.

If the relative position is $\mathbf{d}(t)={\mathbf{d}}_0+t\mathbf{v}$ with $\mathbf{v}\ne 0$, completing the square givesThis proves the minimum distance over all real times. If time is restricted to an interval, project $t_{\ast }$ onto that interval; if $\mathbf{v}=0$, the distance is constant. Both particles are evaluated at the same time, unlike the two independent parameters in the distance between skew lines.