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## Homework Statement

If two particles have velocities u and v in frame S, find their relative speed in frame S'.

## Homework Equations

## The Attempt at a Solution

Isn't it strange that the relative speed doesn't depend on the velocity of the frame, ##\vec s##?

Since the two particles have velocities ##\vec u## and ##\vec v## in some reference frame S, I am to find the relative velocity and speed in frame S'.

Letting the relative velocity of the frame be ## \vec s##, the transformation of velocities are:

[tex]

\vec u' = \frac{1}{1 - \frac{\vec u \cdot \vec s}{c^2}}\left[ \frac{1}{\gamma_s} \vec u - \left( 1 - \frac{\vec u \cdot \vec s }{c^2}\frac{\gamma_s}{1 + \gamma_s} \right) \vec s \right] [/tex]

[tex]

\vec v' = \frac{1}{1 - \frac{\vec v \cdot \vec s}{c^2}}\left[ \frac{1}{\gamma_s} \vec v - \left( 1 - \frac{\vec v \cdot \vec s }{c^2}\frac{\gamma_s}{1 + \gamma_s} \right) \vec s \right] [/tex]

Taking ##\vec u' - \vec v'## only gives me components in ##\vec u, \vec v, \vec s ##. How do I extract the magnitude?