Question

A plane flies 650 mph on a bearing of 175.3°. A 25-mph wind, from a direction of 266.6°, blows against the plane. Find the resulting bearing of the plane.

Accepted Answer

Identify the velocity vectors of the plane and the wind. The plane's velocity vector has a magnitude of 650 mph and a direction (bearing) of 175.3°. The wind's velocity vector has a magnitude of 25 mph and a direction of 266.6° (wind direction means the wind is coming from this bearing, so the wind's velocity vector points toward 266.6° + 180° = 86.6°). Convert both velocity vectors from polar form (magnitude and bearing) to Cartesian coordinates (x and y components) using the formulas:  $$x = v 	imes \sin(	heta)$$ and $$y = v 	imes \cos(	heta)$$, where $$	heta is $$the bearing angle in degrees. Add the corresponding components of the plane's velocity vector and the wind's velocity vector to find the resultant velocity vector components: $$V_x = V_{plane,x} + V_{wind,x}$$ and $$V_y = V_{plane,y} + V_{wind,y}. $$Calculate the magnitude of the resultant velocity vector using the Pythagorean theorem: $$V = \sqrt{V_x^2$ + V_y^2$} ($$this step is optional if only the bearing is required). Find the resulting bearing of the plane by calculating the angle of the resultant vector relative to the north direction using the inverse tangent function: $$	heta = \arctan\left(\frac{V_x}{V_y}ight). $$Adjust the angle to the correct quadrant and convert it to a bearing between 0° and 360°.

A plane flies 650 mph on a bearing of 175.3°. A 25-mph wind, from a direction of 266.6°, blows against the plane. Find the resulting bearing of the plane.

주요 개념

Vector Addition in Navigation

Bearing and Angle Measurement

Trigonometric Components of Vectors