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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?

you've taken v1 and v2 in a wrong way. Instead of addition, you have to subtract the components. use $\displaystyle V_{(w,e)}=V_{(w,p)}+V_{(p,e)}$
where w wind, pplane, e earth and remember directions are important...