Consider two inertial frams O and O', with O' moving along the x axis of O with a constant speed
In O events A and B occur with spacetime coordinates
and
respectively.
The vector in 4 dimensional spacetime from A to B in O is
and the components of velocity in O are
and
The Lorentz transformation fromin O to
in O' is
In O' the components of velocity are
The inverse transformation from O' to O is obtained by changing the sign of
and interchanging
and
and
and
and
and
