boost in standard configuration. If the primed frame is travelling with speed with Lorentz factor in the positive relative to the unprimed frame, then the differentials are Divide the first three equations by the fourth, or which is in which expressions for the primed velocities were obtained using the standard recipe by replacing by and swapping primed and unprimed coordinates. If coordinates are chosen so that all velocities lie in a (common) plane, then velocities may be expressed as (see polar coordinates) and one finds The proof as given is highly formal. There are other more involved proofs that may be more enlightening, such as the one below. General configuration Starting from the expression in coordinates for parallel to the , expressions for the perpendicular and parallel components can be cast in vector form as follows, a trick which also works for Lorentz transformations of other 3d physical quantities originally in set up standard configuration. Introduce the velocity vector in the unprimed frame and in the primed frame, and split them into components parallel (∥) and perpendicular (⊥) to the relative velocity vector (see hide box below) thus then with the usual Cartesian standard basis vectors , set the velocity in the unprimed frame to be which gives, using the results for the standard configuration, where is the dot product. Since these are vector equations, they still have the same form for in any direction. The only difference from the coordinate expressions is that the above expressions refers to vectors, not components. One obtains where is the reciprocal of the Lorentz factor. The ordering of operands in the definition is chosen to coincide with that of the standard configuration from which the formula is derived. Either the parallel or the perpendicular component for each vector needs to be found, since the other component will be eliminated by substitution of the full vectors. The parallel component of can be found by projecting the full vector into the direction of the relative motion and the perpendicular component of {{math|u′}} can be found by the geometric properties of the cross product (see figure above right), In each case, is a unit vector in the direction of relative motion. The expressions for and can be found in the same way. Substituting the parallel component into results in the above equation. Using an identity in and ,These formulae follow from inverting for and applying the