Let a particle be moving at velocity w' in K', and let w'x', w'y', and w'z' denote the (signed) x', y', and z' components of w', respectively. That is, w'2 = w'x'2 + w'y'2 + w'z'2. To obtain the corresponding wx, wy, and wz in K, we take an arbitrary segment of the particle's trajectory in K', say, of Δt', Δx', Δy', and Δz', and calculate the corresponding Δt, Δx, Δy, and Δz values in K:

Δx = Δx' · √1 – v2   / c2    + v · Δt = w'x' · Δt' · √1 – v2   / c2    + v · Δt

Δt = (Δx' · √1 – v2   / c2    · (v / c2) / √1 – v2   / c2    + Δt') / √1 – v2   / c2    = (1 + w'x' · v / c2) · Δt' / √1 – v2   / c2

Δy = Δy' = w'y' · Δt'

Δz = Δz' = w'z' · Δt'

The resulting velocities are:

wx = Δx / Δt = w'x' · (1 – v2 / c2) / (1 + w'x' · v / c2) + v = (w'x' + v) / (1 + w'x' · v / c2)

wy = Δy / Δt = w'y' · √1 – v2   / c2    / (1 + w'x' · v / c2)

wz = Δz / Δt = w'z' · √1 – v2   / c2    / (1 + w'x' · v / c2)

(Theorem) w' < c if and only if w < c.

(Proof) w' < c means that w'y'2 + w'z'2 < c2 – w'x'2. Then w2 = wx2 + wy2 + wz2 < ((w'x' + v)2 + (c2 – w'x'2) · (1 – v2 / c2)) / (1 + w'x' · v / c2)2 = c2. The other direction follows from the interchangeability of K and K'.

This is in line with the earlier established speed limit for particles.

(Theorem) If v → c, then wx → c, wy → 0, and wz → 0.

Whenever v, w'x', w'y', w'z' ≪ c, the equations yield values very close to the classical ones, i.e. wx ≈ w'x' + v, wy ≈ w'y', and wz ≈ w'z'.

The formulas are valid also for the tip of a ray of light, i.e. when w' = c. (The formulas can be derived without making any use of the fact that we are talking about a particle.)

(Theorem) w' = c if and only if w = c.