Vector 1: NW @ 5
NS1 = 5 * cos(315) = 3.54
EW1 = 5 * sin(315) = -3.54
Vector 2: NE @ 10
NS2 = 10 * cos(45) = 7.07
EW2 = 10 * sin(45) = 7.07
TotalNS = NS1 + NS2 = 3.54 + 7.07 = 10.61
TotalEW = EW1 + EW2 = -3.54 + 7.07 = 3.54
Compute resultant vector:
Average Direction = atan(TotalEW / TotalNS) = atan(3.54 / 10.61) = 18.43
Note that I have given all angles in degrees. In practice, you would have to convert to and from radians, since trig functions invariably use radian measure. Also, a standard atan function will produce a quadrantal ambiguity that you will have to resolve yourself. Some Microsoft products have an ATAN2 function which takes the rise and run separately, and can therefore resolve the ambiguity itself. When I did this example in Excel, I used ATAN2(10.61, 3.54).
To be mathematically rigorous, you should divide the TotalNS and TotalEW by 2, since you want an average, not a total. However, if you are only interested in the direction, you would be dividing both the numerator and denominator by the same amount, which is an unnecessary step. The true vector average (with the division by 2) would only be necessary if you wanted a magnitude to go along with the direction. In the case of wind, the result would be meaningless.
If the atan function returns a negative result, add 360 degrees to get a compass direction.