An airplane has a ground speed of 523 mph and a component form of the velocity of <-482.781, -39.262>, due to wind. What is the bearing of the plane?

To find the bearing of the plane, we need to use the concept of vector addition. The ground speed represents the resultant vector of two components: the plane's velocity relative to the air and the wind velocity.

The velocity of the plane relative to the air is given as <-482.781, -39.262>. This means that, without any wind, the plane would be moving in the opposite direction of these vector components.

To find the wind's velocity, we need to subtract the plane's velocity relative to the air from the total ground speed. Let's denote the wind velocity as <x, y>.

To find x, we subtract the x-components of the two vectors:
523 = -482.781 + x
x = 523 + 482.781 = 1005.781

Similarly, to find y, we subtract the y-components:
0 = -39.262 + y
y = 39.262

Therefore, the wind velocity is <1005.781, 39.262>.

Now, we have the velocity of the plane relative to the air and the wind velocity. To find the bearing, we need to determine the angle between the ground speed vector and the reference direction (usually north).

Using trigonometry, we can find the bearing angle. The bearing angle can be calculated as the inverse tangent (arctan) of the ratio of the y-component to the x-component of the ground speed vector.

Bearing = arctan(y-component / x-component) = arctan(-39.262 / 1005.781)

To find the bearing angle in degrees, we can use a calculator to find the arctan value and convert it to degrees.

Bearing ≈ -2.24 degrees

Therefore, the bearing of the plane is approximately -2.24 degrees.