Physics (simple trig): a novice pilot sets a plane's controls, thinking that the plane will fly 250 km/hr to the north. if the wind blows at 75 km/hr to the south east (135 degrees), what is the plane's resultant velocity?

The velocity of the plane, the wind vector and the resultant plane velocity form a triangle.
P = velocity vector of plane
W = wind vector
  V = resultant plane velocity
   P + W = V
You can calculate vector components of P and W and combine them to get V.
 Px = 0 Wx = 75cos(45) = (75/2)SQRT(2)
 Py = 250 Wy = -75sin(45) = -(75/2)SQRT(2)
 Vx = Px + Wx = 0 + (75/2)SQRT(2) = (75/2)SQRT(2) = 53.033
 Vy = Py + Wy = 250 - (75/2)SQRT(2) = 196.970
 A = Angle from north of resultant velocity
 tan(A) = Vx/Vy = 53.033/196.970 = 0.269248
 A = 15.07 degrees
v = Magnitude of resultant velocity = SQRT{(53.033)^2 + (196.970)^2]
v = 203 km/h
 The resultant velocity is 203 km/h at an an angle of 15 degrees east of north.