  Using spacetime algebra, the geometric algebra of spacetime, the general
problem of relativistic addition of velocities is addressed. The successive
application of non-collinear Lorentz boosts is then studied in Minkowski
spacetime. Even spatial vectors, such as the relative velocity of two reference
frames, are analyzed in their proper setting: as vectors belonging to a
four-dimensional spacetime manifold and not as vectors in ordinary
three-dimensional (Euclidean) space. This is a clear and physical result that
illustrates the fact that Lorentz boosts do not form a group. The entire
derivation is carried through without using the second postulate on the speed
of light, thereby stressing that the framework of special relativity does not
depend on electromagnetism.
