An airplane is 50,000 m above an observer and 2.1 km to the west of them and 1.5 km to the north of you. Determine the angle to the plane in the x – y axis and the total distance to the plane from you. Choose the x-axis east, y axis north, and z axis up.
To determine the angle to the plane in the x-y axis and the total distance to the plane, we can use the Pythagorean theorem and trigonometric functions.
First, let's determine the distance between you and the plane. We can calculate this using the Pythagorean theorem:
Distance^2 = (Distance in x-axis)^2 + (Distance in y-axis)^2
Given that the plane is 2.1 km to the west of you and 1.5 km to the north of you, we can substitute the values into the formula:
Distance^2 = (2.1 km)^2 + (1.5 km)^2
Distance^2 = 4.41 km^2 + 2.25 km^2
Distance^2 = 6.66 km^2
Taking the square root of both sides, we get:
Distance = √(6.66 km^2)
Distance ≈ 2.58 km
So the total distance to the plane from you is approximately 2.58 km.
Next, let's determine the angle to the plane in the x-y axis. We can use trigonometric functions to calculate this angle.
The angle in the x-y plane (measured counterclockwise from the positive x-axis) can be calculated using the inverse tangent function (arctan).
Angle = arctan((Distance in y-axis) / (Distance in x-axis))
Given that the plane is 2.1 km to the west of you and 1.5 km to the north of you, we can substitute the values into the formula:
Angle = arctan(1.5 km / 2.1 km)
Angle ≈ arctan(0.714)
Using a calculator, we can find that Angle ≈ 35.97 degrees.
Therefore, the angle to the plane in the x-y axis is approximately 35.97 degrees, and the total distance to the plane from you is approximately 2.58 km.