Final answer:
The speed and direction of the wind can be determined using vector addition by comparing the airplane's velocity relative to the air with its velocity relative to the ground.
Step-by-step explanation:
To determine the speed and direction of the wind affecting the airplane, we must use vector addition. The airplane's velocity relative to the air is 45.0 m/s due north, while its velocity relative to the ground is 38.0 m/s at an angle west of north. The difference between these two vector quantities will give us the wind's vector.
We can represent the airplane's velocity as a vector pointing straight up (north) on a graph, which is 45.0 m/s. The velocity relative to the ground can be represented as a vector at an angle to the first vector, with a magnitude of 38.0 m/s. To find the wind's velocity, we draw the wind's vector as the third side of a triangle formed by these two vectors. Using trigonometry, especially the Pythagorean theorem and sine and cosine functions, we can determine the wind's magnitude and angle relative to the north.
The resulting vector, pointing west of north and with a smaller magnitude than the airplane's airspeed, demonstrates that the wind has both a crosswind component (from the east) and a head-wind component (blowing against the northward direction of the plane), hence reducing its ground speed and changing its course.