An airplane is 5,000 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, we can use trigonometry. Let's break down the given information:
- The airplane is 5,000 m above the observer, which is the same as saying it is 5 km above.
- The airplane is 2.1 km to the west of the observer, and 1.5 km to the north of you.
First, let's calculate the distance between you and the observer. This can be done using the Pythagorean theorem:
Distance = √((2.1 km)^2 + (1.5 km)^2)
Distance = √(4.41 + 2.25)
Distance = √6.66
Distance ≈ 2.58 km
Now, we can calculate the total distance from you to the plane. Again, we use the Pythagorean theorem:
Total Distance = √((Distance)^2 + (5 km)^2)
Total Distance = √((2.58 km)^2 + (5 km)^2)
Total Distance = √(6.6564 km^2 + 25 km^2)
Total Distance = √(31.6564 km^2)
Total Distance ≈ 5.63 km
To determine the angle to the plane in the x-y axis, we can use trigonometry:
Angle = atan(1.5 km / 2.1 km)
Angle ≈ 34.4 degrees
So, the angle to the plane in the x-y axis is approximately 34.4 degrees, and the total distance to the plane from you is approximately 5.63 km.
To determine the angle to the plane in the x-y axis and the total distance to the plane from you, we can use trigonometry and Pythagorean theorem.
First, let's define our reference point as the observer. The observer is at the coordinates (0,0,0) in the x-y-z coordinate system.
Now, let's consider the position of the airplane. It is 2.1 km to the west of the observer (negative x-direction) and 1.5 km to the north of the observer (positive y-direction). The airplane is also 5,000 m above the observer (positive z-direction).
To find the angle to the plane in the x-y axis, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is 1.5 km (y-coordinate of the airplane) and the adjacent side is 2.1 km (negative x-coordinate of the airplane).
The angle to the plane in the x-y axis can be found as:
angle = atan(opposite/adjacent)
= atan(1.5/2.1)
Using a calculator, we can find the angle to be approximately 35.36 degrees.
Next, to find the total distance to the plane from the observer, we can use the Pythagorean theorem. The distance is the hypotenuse of a right triangle formed by the x, y, and z coordinates of the airplane relative to the observer.
The total distance can be found as:
distance = sqrt(x^2 + y^2 + z^2)
= sqrt((-2.1)^2 + (1.5)^2 + (5)^2)
Using a calculator, we can find the total distance to be approximately 5.97 km.
Therefore, the angle to the plane in the x-y axis is approximately 35.36 degrees, and the total distance to the plane from the observer is approximately 5.97 km.