How is the Final Velocity of Star Movements Calculated?

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SUMMARY

The final velocity of star movements is calculated by combining radial and proper (tangential) motions using vector addition. For example, Barnard's Star has a tangential velocity of 90 km/s and a radial velocity of 111 km/s, resulting in a space velocity of approximately 142 km/s when applying the Pythagorean theorem. However, this method does not account for relativistic effects, such as those described by the transverse Doppler shift, which should be considered for more accurate calculations in high-velocity scenarios.

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  • Understanding of vector addition in physics
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  • Knowledge of the Pythagorean theorem
  • Basic principles of relativistic physics
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When calculating the space velocities of stars by combining what we see of both their radial and proper (tangential) motions, how is the final velocity determined?I don't believe they just add these two vectors together do they? Do they make use of the transverse Doppler shift calculation here?
 
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For instance, apparently Bernard star has a tangential velocity of 90 and a radial of 111. Using plain pythagorans, this works out to 142 in space velocity. (see http://en.m.wikipedia.org/wiki/Proper_motion?wasRedirected=true)

why are these two vectors just added together so simply like this when there ought to be a relativistic component to it as used in the transverse Doppler shift calc?
 

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