difference of two squares to obtain so that where the last expression is by the standard vector analysis formula . The first expression extends to any number of spatial dimensions, but the cross product is defined in three dimensions only. The objects with having velocity relative to and having velocity relative to can be anything. In particular, they can be three frames, or they could be the laboratory, a decaying particle and one of the decay products of the decaying particle. Properties The relativistic addition of 3-velocities is non-linear, so in general for real number , although it is true that Also, due to the last terms, is in general neither commutative nor associative It deserves special mention that if and refer to velocities of pairwise parallel frames (primed parallel to unprimed and doubly primed parallel to primed), then, according to Einstein's velocity reciprocity principle, the unprimed frame moves with velocity relative to the primed frame, and the primed frame moves with velocity relative to the doubly primed frame hence is the velocity of the unprimed frame relative to the doubly primed frame, and one might expect to have by naive application of the reciprocity principle. This does not hold, though the magnitudes are equal. The unprimed and doubly primed frames are not parallel, but related through a rotation. This is related to the phenomenon of Thomas precession, and is not dealt with further here. The norms are given by and It is clear that the non-commutativity manifests itself as an additional rotation of the coordinate frame when two boosts are involved, since the norm squared is the same for both orders of boosts. The gamma factors for the combined velocities are computed as Notational conventions Notations and conventions for the velocity addition vary from author to author. Different symbols may be used for the operation, or for the velocities involved, and the operands may be switched for the same expression, or the symbols may be switched for the same velocity. A completely separate symbol may also be used for the transformed velocity, rather than the prime used here. Since the velocity addition is non-commutative, one cannot switch the operands or symbols without changing the result. Examples of alternative notation include: No specific operand (using units where c = 1) Left-to-right ordering of operands Right-to-left ordering of operands Applications Some classical applications of velocity-addition formulas, to the Doppler shift, to