adopt a different convention. Continuing to work in the (non-relativistic) Newtonian limit we begin with a Galilean transformation in one dimension: where x' is the position as seen by a reference frame that is moving at speed, v, in the "unprimed" (x) reference frame. Taking the differential of the first of the two equations above, we have, , and what may seem like the obvious statement that , we have: To recover the previous expressions for relative velocity, we assume that particle A is following the path defined by dx/dt in the unprimed reference (and hence dx′/dt′ in the primed frame). Thus and , where and refer to motion of A as seen by an observer in the unprimed and primed frame, respectively. Recall that v is the motion of a stationary object in the primed frame, as seen from the unprimed frame. Thus we have , and: where the latter form has the desired (easily learned) symmetry. Special relativity As in classical mechanics, in Special Relativity the relative velocity is the velocity of an object or observer B in the rest frame of another object or observer A. However, unlike the case of classical mechanics, in Special Relativity, it is generally not the case that This peculiar lack of symmetry is related to Thomas precession and the fact that two successive Lorentz transformations rotate the coordinate system. This rotation has no effect on the magnitude of a vector, and hence relative speed is symmetrical. Parallel velocities In the case where two objects are traveling in parallel directions, the relativistic formula for relative velocity is similar in form to the formula for addition of relativistic velocities. The relative speed is given by the formula: Perpendicular velocities In the case where two objects are traveling in perpendicular directions, the relativistic relative velocity is given by the formula: where The relative speed is given by the formula General case The general formula for the relative velocity of an object or observer B in the rest frame of another object or observer A is given by the formula: where The relative speed is given by the formula See also Doppler effect Peculiar velocity Proper motion Range rate Radial velocity Rapidity Relativistic speed Space velocity (astronomy) Notes References Further reading Alonso & Finn, Fundamental University Physics Greenwood, Donald T, Principles of Dynamics. Goodman and Warner, Dynamics. Beer and Johnston, Statics and Dynamics. McGraw Hill