This Demonstration considers the composition of velocities in accordance with the special theory of relativity. Consider a system moving with velocity represented by the red arrow, with magnitude

and direction

, with respect to a stationary frame of reference. The red disk recapitulates this magnitude, which has an upper limit extending to the red circle, corresponding to the speed of light

. The blue arrow represents a second velocity, which has a magnitude

and direction

, with respect to the moving frame of reference. The velocity with respect to the original stationary frame is then represented by

. A compact formulation gives the components of

parallel and perpendicular to

:

The gray arrow represents the vector

.