A plane flying with an airspeed of $540$ mph in the direction $\text{N }45^\circ\text{ E}$ ($45°$east of north) encounters a wind blowing at $47$ mph in the direction $\text{S }78^\circ\text{ E}$.  

1. What are the true speed and direction of the plane?

   Step (a):

   $\left\langle540\cos\left(45^\circ\right)+47\cos\left(-12^\circ\right),540\sin\left(45^\circ\right)+47\sin\left(-12^\circ\right)\right\rangle$

2. How long will it take the plane to travel $1000$ miles in the resultant direction?

   Step (b):

   Solve (magnitude of vector in part (a)) $t=1000$.

3. If the pilot intended to fly $1000$ miles in the direction $\text{N }45^\circ\text{ E}$, how far from his intended destination is he given the situation in part (b)?

   Step 1 (c):

   intended distance vector: $\left\langle1000\cos(45^\circ),1000\sin(45^\circ)\right\rangle$

   Step 2 (c):

   actual distance vector: $\left\langle(540\cos(45^\circ)+47\cos(-12^\circ))t,540\sin(45^\circ)+47\sin(-12^\circ))t\right\rangle$
   Use the time you calculated in part (b).

   Step 3 (c):

   Determine the magnitude of the difference of the vectors in steps 1 and 2.