# Application of vectors in problem solving

An aircraft cruises at a speed 300 km/h in still air. If the wind is blowing from the east at 100 km/h, in what direction should the aircraft head in order to fly in a straight line from city P to city Q, 400 km north-northeast of P? How long will the trip take?

The magnitude of the green vector is what I wan't to figure out, and the angle v created by the red vector (v is theta).

How do I solve this?

First, draw an appropriate picture that represents the situation at hand.

Then solve the following equations for $$\theta$$ and ground speed $$S$$. Let $$\alpha$$ be the angle of $$22{1\over 2}^\circ$$.

$$\sin \theta - S {\sin \alpha \over 300} = {100 \over 300}$$, $$\cos \theta - S {\cos \alpha \over 300} = 0$$.

Explicitly:

Multiply the first by $$\cos \alpha$$ and the second by $$- \sin \alpha$$ and add the equations to get $$\sin ( \theta-\alpha) = {100 \over 300} \cos \alpha$$.

Computing numbers:

Substituting numbers gives $$\theta \approx 40.4^\circ$$ and $$S \approx 247$$ km/hr.

Given the ground speed the time taken can be easily computed.