# Airplane ground speed and direction, given airspeed and wind

An airplane heads northeast at an airspeed of 900 km/hr, but there is a wind blowing from the west at 60 km/hr. In what direction does the plane end up flying? (angle). What is its speed relative to the ground?

I drew a picture. I think I am having a hard time understanding how to solve the problem. I want to add the wind vector to the plane vector - is that as simple as 840 km/hr? After really adding the vectors together, do I do: (added vectors)cos(45)?

• Have you drawn a picture yet? It would help if you broke the velocity vector into pieces: what amount is north/south? What amount is east/west? – NoseKnowsAll Sep 9 '15 at 17:40
• @Jake the two vectors make an angle of 135 so use law of cosines and sines to find the required. – user258250 Sep 9 '15 at 17:50
• @NoseKnowsAll Yes I drew a picture. I think I am having a hard time understanding how to solve the problem. I want to add the wind vector to the plane vector - is that as simple as 840 km/hr? After really adding the vectors together, do I do: (added vectors)cos(45) thanks – Jake Mager Sep 9 '15 at 18:01

The velocity of the plane relative to the wind is $$_P\underline{v}_W=\underline{v}_P-\underline{v}_W$$
Therefore, $$\frac{900}{\sqrt{2}}\left(\begin{matrix}1\\1\end{matrix}\right)=\underline{v}_P-\left(\begin{matrix}60\\0\end{matrix}\right)$$
So now you can find the velocity of the plane $\underline{v}_P$.Then you can find the direction and magnitude.