Trig question on bearing of a plane using vectors

Question

A plane is heading due south with an airspeed of 211 mph. A wind from a direction of 53 degrees is blowing at 17 mph. Find the bearing of the plane.

Answer 1

Draw a picture

I assume that "heading due south" is the motion of the plane without wind. We will then add the wind vector to get the actual motion of the plane with respect to the ground.

Put due north at the top of the drawing---therefore, the airspeed vector is pointing straight down, with a length of 211. The wind is specified as "coming from" a direction of 53 degrees. This means it is pointing down and to the left. sketch it in with the magnitude at the appropriate scale.

You will see that the sum of the 2 vectors is just a bit larger than 211, and that the direction is a small angle to the left of straight down.

The math:

We know the magnitude and direction of the wind, so find the horizontal and vertical components:

horizontal = H = 17sin(53) (To the left---AKA west)

and vertical = V = 17cos(53) (Down---AKA south)

To the vertical, add the airspeed vector to get V(total) =211 + 17cos53

You now have the horizontal a vertical components of the motion with respect to the ground. The bearing of the net motion is the inverse tangent of H / V(total)