Intuitively we add or subtract velocities along the same line to obtain relative velocities.
The velocity of the object in frame O' is
According to the Galilean transformation, the velocity in frame O is
If
then
twice the speed of light. We know in fact that nothing can travel faster than the speed of light, so the Galilean transformation is wrong. Special relativity predicts that velocities be added using the formula
Using this withgives
The frames above are separating, but we can use that velocity is a vector to deal with the case of approaching inertial frames.
If
and
then