Angle of star proper movement

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

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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