"Adding unit proper-velocities": In this case object A is traveling in various directions with respect to object B as shown by the red "w"AB vectors in the figure above, while object B is traveling east with respect to object C as shown by the green "w"BC vectors in the figure above. This illustrates how vector-addition can help us figure out the proper velocity of object A with respect to object C, as shown by the blue "w"AC vectors in the figure above. Note that when adding proper-velocities (i.e. map-distance r per unit traveler-time τ) at high speed e.g. to get "w"AC from "w"AB and "w"BC, the out-of-frame component (here A with respect to B) must be rescaled (orange extension) to become ("w"AB)C by a factor ≥ 1 before the vector-addition is performed. At high speeds, this magnitude-rescaling is needed in general (even in the uni-directional case).