 So let's suppose Bob is standing here in his frame and he sees a rock go flying past. Now he measures the speed of the rock to be ux,b. So the x is to tell us it's in the x direction and b means it's Bob's measurement. Now Alice is in her frame here traveling with some speed v. What does Alice measure this speed to be? What is ux,a? So speed is distance divided by time or delta x over delta t. So in Alice's frame we have ux,a is delta x,a over delta t,a. So here we substitute in the Lorentz transformations. So this doesn't look very helpful, but now we divide the top and bottom of the fraction by delta t,b. And we get this. Now delta x,b on delta t,b is ux,b, the speed of the ball in Bob's frame. And so we arrive at this. Now let's have a check if this is correct. What happens in everyday life if the velocities involved are much smaller than c? Well in this case the denominator reduces to one and we have ux,b minus v, which is what you'd expect. If you were running away from it at speed v, you'd see it's being reduced by v. So long as you're not traveling fast enough to see relativistic effects. So what we call this is the longitudinal velocity subtraction formula. So what do I mean by that? Longitudinal just means Alice's velocity is in the same axis as the rock's velocity. They're both along the x-axis. And it's velocity subtraction because Alice is moving away from it. Now what about the velocity addition case? If Alice is running towards the rock. Well in this case the derivation is exactly the same except now v gets replaced with minus v. Finally what about the transverse velocity? This means the rock is moving at 90 degrees to Alice's motion. So we need to add the Lorentz transformation for y and speed is distance divided by time. Delta y over delta ta. So just following along with what we did earlier, what does this give us?