Using rapidity - prove velocity transformation equations

[zimo]

Homework Statement

Prove the law of transformation of velocities

$$\begin{array}{l}<br /> {v_x} = \frac{{{v_x}^\prime + V}}{{1 + {v_x}^\prime V/{c^2}}}\\<br /> {v_y} = \frac{{{v_y}^\prime }}{{\gamma (1 + {v_x}^\prime V/{c^2})}}\\<br /> {v_z} = \frac{{{v_z}^\prime }}{{\gamma (1 + {v_x}^\prime V/{c^2})}}<br /> \end{array}$$

Homework Equations

Rapidity equations

The Attempt at a Solution

I proved Vx with an ease using rapidity equations and the identity for tanh(a+b)
now, I'm stuck with revealing the relation to Vy and Vz.

 

[Rasalhague]

What happens if you use the same method you used to prove the first equation, except taking into account that a distance in some direction orthogonal to V is the same in both coordinate systems?