An airplane is flying in the direction of 148°, with an airspeed of 875 kmh. Because of the wind, its groundspeed and direction are 800 kmh and 140°, respectively. Find direction and speed of the wind.

the airplane is heading South east.

My set up:

v1 : 875 < cos148, sin148 >
v2 : 800 < cos140, sin140 >

v= v1 + v2
v= -1354.93, 980.91

Resultant speed, ||V|| = 1672.73

Direction : tan inverse ( 980.91/ -1354.93) = -35.90°

The answer they got is :

N 21.4° E; 138.7 kmh.

Am I doing this completely wrong?

2. you've taken v1 and v2 in a wrong way. Instead of addition, you have to subtract the components. use $V_{(w,e)}=V_{(w,p)}+V_{(p,e)}$

where w- wind, p-plane, e- earth and remember directions are important...