A star lying 45 pc from the Sun has proper motion of 0.2 "/yr. What is the star'

You can use this equation for the transverse velocity:

v(transverse) = (45 pc) tan (0.2") per year = 42.66 km/sec

Actually, the following equation is better, but the angle is so small that it doesn't make a difference:

v(transverse) = (45 pc) (2) sin (0.1") per year = 42.66 km/sec

For the radial velocity, I'll use the relativistic doppler shift equation.

Lo / Le = sqrt { (1 + v/c) / (1 - v/c) } = 1.0002
1 + v/c = (1.0002)^2 (1 - v/c)
1 + v/c = 1.00040004 - 1.00040004 v/c
2.00040004 v/c = 0.00040004
v(radial) = 0.00019998 c = 59.95 km/sec

Then the Pythagorean theorem.

v = sqrt{ [v(radial)]^2 + [v(transverse)]^2 } = 73.58 km/sec