Answer:
195.2 mph at 337.3°
Step-by-step explanation:
You want the speed and direction of the wind that causes a plane flying at 330 mph on a bearing of 65° to have a true speed and direction of 390 mph on a bearing of 35°.
Difference
The wind vector is the difference between the plane's true course vector and its nominal course.
wind = 390∠35° -330∠65° = 195.2∠337.3°
The wind is 195.2 mph in the direction 337.3°.
Rectangular coordinates
In (north, east) coordinates, the computation is ...
wind = 390(cos(35°), sin(35°)) -330(cos(65°), sin(65°))
= (319.4693, 223.6948) -(139.4640, 299.0816) = (180.0053, -75.3868)
Then the speed is ...
speed = √(180.0053² +(-75.3868)²) ≈ √38085.0776 ≈ 195.154
and the direction is ...
arctan(-75.3868/180.0053) ≈ -22.72° = 337.28°
The wind velocity is 195.2 mph at 337.3°.
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Additional comment
Bearing angles are measured clockwise from north (+y), so it is convenient to use (north, east) rectangular coordinates for the math. That way, we avoid the extra translation of angles to/from CCW from +x.
The angle -22.7° is N22.7°W. Measured CW from north, that is 337.3°.
