Bernard's Star is 5.963 light years away with a proper motion of 10.33 arc seconds per year and a hydrogen alpha line measured at 656.047 nm. What...

There are two components of the motion here: The proper motion, which we see as horizontal because it is moves the star neither closer nor further relative to us, and the redshift, which is due to the expansion of the universe and also the star's inherent movement toward or away from us. These are two perpendicular components of a vector that comprises the star's velocity relative to our solar system.

Let's start with the proper motion. For a star at distance `r` away from us with a proper motion in radians of `{d theta} / {d t}`the horizontal velocity is:
`v = d/dt[r tan theta] = r sec^2 theta {d theta}/{d t} approx r {d theta}/{d t}`
Since the angles are so small (arcseconds), we don't actually need to worry about the secant term, which is very close to 1.

The important thing is to convert to radians; there are 3600 arcseconds in a degree, and `180/pi` degrees in a radian.

`10.33 {arcsec}/{yr} * ({1 deg}/{3600 arcsec})({pi rad}/{180 deg}) = 5.01*10^-5 {rad}/{yr}`
Then just multiply by the distance:
`v = r {d theta}/{dt} = (5.963 light-yr)(5.01*10^-5 {rad}/{yr}) = 2.97*10^-4 {light-yr}/{yr}`
Now we need to figure out the redshift. The hydrogen alpha line has redshifted to 656.047 nm, but what was it originally? We actually need quite a precise answer in this case. The most precise value I've been able to look up is 656.281 nm. Our the observed wavelength is shorter, indicating blueshift; so this star is actually moving toward us, against the expansion of the universe.

To find its velocity, we use the Doppler shift formula:
`lambda_1/lambda_0 = sqrt{{v+c}/{v-c}}`
For velocities this small, we can use an approximation instead:
`v/c approx {lambda_1 - lambda_0}/{lambda_0}`
Where a positive value for `v` indicates redshift, and a negative value indicates blueshift. Substitute in, recalling that by definition `c = 1 {light-yr}/{yr}` :
`v = c {656.047 nm - 656.281 nm}/{656.281 nm} = -3.57*10^-4 {light-yr}/{yr}`

Those are each of our components. To get our overall velocity and angle of approach, we just use the standard formulas for vector components:
`v = sqrt{v_x^2 + v_y^2} = sqrt{(2.97*10^-4)^2 + (-3.57*10^-4)^2} = 4.64*10^-4 {light-yr}/{yr}`
Call the angle `phi` ; note that this angle is not negligibly small:
`phi = tan^{-1} (v_y / v_x) = tan^{-1} ({2.97*10^-4}/{3.57*10^-4}) = tan^{-1} (0.832) = 39.8 deg ` This star is approaching us at a speed of 4.64*10^-4 light-years per year, at an angle of 39.8 degrees from head-on. Since it is 5.963 light-years away, if nothing else changed, it would go past us in a little over 10,000 years.