Aplane's airspeed is 400 miles per hour on a heading of 45 degrees south of east (compass: 135 degrees, babylonian 315 degrees). if a wind is blowing from the west with a speed of 50 miles per hour, what is the ground speed and direction of the plane?

If we are to draw the motion of the plane and the wind in a Cartesian plane, we take East and North as the positive x and y-axis, respectively. Also, we have the West and South as the negative x- and y-axis, respectively. 

To answer this item, we take the x- and y-components of the speeds. 

Airplane:
      x-component = (400 mi/hr)(cos 45°) = 282.84 mi/h
       y-component = -(400 mi/hr)(sin 45°) = -282.84 mi/h

Wind:
     x-component = -(50 mi/hr)(sin 90°) = -50 mi/hr
     y-component = 0 mi/hr

Adding up the components:
     x-component = 282.84 mi/hr - 50 mi/hr = 232.84 mi/hr
     y-component = -282.84 mi/hr

The resultant speed would be,
     R = sqrt ((232.84 mi/hr)² + (-282.84 mi/hr)²) = 366.35 mi/h

The direction would be,
     tan (∅) = -282.84 mi/hr / 232.84 mi/hr 
      ∅ = 50.53° (south of east)

ANSWER: 366.35 mi/hr, 50.53° South of East