There's something I don't understand about that type of explanation.

Say a particle and I are moving along like so:

    -----[photon @ speed of light]-->
    -----[me @ 50% speed of light]-->

Or more simplified, say two cars are moving like so:

    -----[other car]-->
    -----[my car]-->

When I'm stationary, I see him go past at 100km/h. When I move at 50km/h, in the real world he would now appear to only be going 50km/h. But if I slow down my time to half speed, now he'll look like he's going 100km/h again. Because he looks to me like he's at 50km/h at double time speed = 100km/h. That seems to make sense - time slows down and the apparent speed stays at C (100km/h in this case).

That's how I've understood the explanation for relativity. But it must be a very wrong understanding - because that whole idea breaks down when the other car is going along a different vector to me! In this situation:

    <----[other car @ 100km/h]--
    -----[my car @ 50km/h]-->

His speed relative to me would come out as 150km/h in the real world, and no amount of making him look like he's going faster would reduce his speed to 100km/h.