Okay, I think I made a mistake in writing about proper motion, it is the motion projected on the celestial sphere, with a nice
descriptive image here.
But you seem to mix up things. RA and Dec describe the location of a star on the celestial sphere
with a descripitve image here. So, to be clear, you have to at least talk about the RA-velocity or time-variation, in oder to get the proper motion of the star. Also, with RA and Dec being angles, you cannot just do a pythagorean addition. Naturally, when the proper motion is small, then sin(a) = a etc. and then you can come to some sort of equation like you wrote down.
But then, I don't understand how you want to translate that from Vega to Canopus.
But reading again your OP, it looks like you already have the proper motion of both stars, so on the celestial sphere you can easily plot those vectors. If you want then to compare the two, you just take a projection of the northern and southern polar region and plot the proper motion vector in RA-Dec space.
To use the images that you linked to, which are spherical coordinates in some way, you will have to figure out how the coordinate system of the figs compares to RA-Dec and then make the appropriate correction (which I cannot do at the moment here on the couch at home).