# Does Einstein velocity addition rule work only for speeds, or for velocity vectors as well?

Einstein's Velocity addition rule (https://en.wikipedia.org/wiki/Velocity-addition_formula#Special_relativity) was used to describe to replace galileo's in account for relativity. However, in many textbook, the derivation was done only at $$x$$ direction, and the notion was somewhat confusing.

Does Einstein's velocity addition work for vector, or just the speed?

i.e. was $$\displaystyle \vec{v}_{AC}=\frac{\vec{v}_{AB}+\vec{v}_{BC}}{1+(\vec{v}_{AB}\cdot \vec{v}_{BC}/c^2)}$$ true, or only $$\displaystyle v_{AC}=\frac{v_{AB}+v_{BC}}{1+(v_{AB}v_{BC}/c^2)}$$(where $$v_{AB}$$ and $$v_{BC}$$ were both in the same direction)?

Further, if it's true for vector, how to prove it? If not, if there any way to extend it to vector?

Mohammed is right, the general form is more complicated. The reason is that in SR the relativistic effects only apply to the components of velocity in the direction of the relative motion of the reference frames involved- components that are normal to the direction of motion are unaffected, so that has to be taken into account when the velocities and reference frames are arbitrarily aligned with each other. The typical explanations of the Lorenz transformations presented in textbooks consider a special case in which motion is aligned along the x axis of one reference frame which in turn is aligned along the x axis of another, as that simplifies the mathematics while still making the principle clear.

No. The general form of the velocity addition formula is complicated and can be found in your mentioned link (See the blue box.). The general form is neither commutative nor associative.

You can solve the general problem for V1 and V2 by considering Va and Vb where;

Va = (V1 + V2)/2 and Vb = (V1 - V2)/2.

V1 is then Va + Vb and V2 is Va - Vb

Depending on how your problem is presented you end up adding two equal velocities in the same direction.

V1 &+ V2 is calculated from Va &+ Va

V1 &- V2 is calculated from Vb &+ Vb

Where &+ is relativistic addition, and &- is relativistically 'seen from'