An airplane is flying 340 km/hr at 12o east of north. the wind is blowing 40 km/hr at 34o south of east. what is the plane's actual velocity?

Define an x-y coordinate system such that
The positive x-axis = the eastern direction, with unit vector  .
The positive y-axis = the northern direction, with unit vector .

The airplane flies at 340 km/h at 12° east of north. Its velocity vector is


The wind blows at 40 km/h in the direction 34° south of east. Its velocity vector is


The plane's actual velocity is the vector sum of the two velocities. It is


The magnitude of the actual velocity is
v = √(121.1615² + 306.0473²) = 329.158 km/h

The angle that the velocity makes north of east is
tan⁻¹ (306.04733/121.1615) = 21.6°

The actual velocity is 329.2 km/h at 21.6° north of east.