Figure out the location of celestial objects relative to the Milky Way in x, y, z co-ordinates

The spherical $(r, l, b)$ can be converted to rectangular $(U, V, W)$ as:

$$begin{align} U &= r cos b cos l \ V &= r cos b sin l \ W &= r sin b end{align}$$

Using $(r, l, b) = (770 pm 40 ~text{kpc}, 121.2^circ, -21.6^circ)$,

$$begin{align} U &= -371 mp 19 ~text{kpc} \ V &= ~~~ 612 pm 32 ~text{kpc} \ W &= -283 mp 15 ~text{kpc} end{align}$$

That's relative to the Sun. For a position relative to the center of our galaxy, subtract $8 pm 0.5 ~text{kpc}$ from $U$.