Transforming velocity vector from its one coordinates system to another coordinate

Question

If I have a velocity of an object in its local coordinates, and I have the velocity of another object

how would I transform the velocity vector to the coordinates of the second object.

Answer 1

That simple looking question touches an amazing amount of physics.

If the velocities are small compared to the speed of light, then have a look at Galilean transformations.

Otherwise it gets more complicated, see Lorentz transformations, Poincaré transformations.