An airplane is traveling at fixed altitude with speed of 500 mph in a path that makes an angle of 120 degrees with positive x-axis. At a certain point, the airplane encounters wind with velocity of 70 mph in direction that is 45 degrees North East.
What are the resultant speed and direction of the plane?
Okay so I partially understand the question and here is my step by step process.
1) I let $$\begin{align} \mathbf v_1 &= 500\cos(120^\circ)\mathbf{i} + 500\sin(120^\circ)\mathbf{j} \\ \mathbf v_2 &= 70\cos(45^\circ)\mathbf{i} + 70\sin(45^\circ)\mathbf{j} \end{align} $$ Okay this is as far as I got.. and I am stuck.. I really don't know how to go from here. Mainly because I got confused with how I should evaluate $v$... I know there is a relationship between $\mathbf v$, $\mathbf v_1$ and $\mathbf v_2$ in terms of vectors but the way I visualized it at first was to try and determine the inner angles but I got nowhere with that and then I considered using the magnitudes of $\mathbf v_1$ and $\mathbf v_2$ to determine v but that went nowhere as well..