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I'm trying to figure out how to calculate the heading and flight path of an aircraft when it's measured from the ground. I literally have no idea where to start.

The coordinates from the ground are (-5.60, -20.91, 24.14) and the aircraft's speed is 374.16 n mi/hr

I'm really just looking for a formula or directions to a good resource, not looking for the problem to be solved.

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  • $\begingroup$ One point and speed does not make a vector. So an answer is impossible. $\endgroup$ – StainlessSteelRat Feb 8 at 14:05
  • $\begingroup$ The answer isn't impossible, so there is that. But thanks for taking a look. $\endgroup$ – David Scidmore Feb 9 at 2:35
  • $\begingroup$ You need two points to get a direction. Two points over a fixed time will give you speed and direction. Speed and one point is not enough. $\endgroup$ – StainlessSteelRat Feb 9 at 3:33
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To determine a heading and path, also known as a vector, you have to have a starting point, which you may have (see below), a speed, which you have (374.16 kts), but also a direction. The direction is calculated, as you request, by applying appropriate formulae.

The math becomes involved once the plotting moves from 2-space (flat plane, not aeroplane) to 3-space and the formulae can be found online in many locations. One such location, Interactive Mathematics, provides as clearly as possible on the flat computer screen how to think about these vectors and provides the math.

vector diagram

If you consider that your origin point is 0, 0, 0 and the points you've provided are represented in the drawing above, you have a good portion of the information required.

I've enjoyed vector math in the past, but that was more than 30 years ago and there aren't enough brain cells to get beyond this point for me.

I suspect there's a mathematics SE better suited for your question.

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Let's use the aviation words:

  • Heading is the direction the airplane is pointed to.

  • Track is the actual direction of the airplane tracking across the ground

  • Course is your intended path of travel to your destination.

A pilot tries to track close to course even if he may have to fly in a different heading to counter the effect of the wind.

The information you have is not enough to calculate the track of the flight (if we interpret flight path as a track), nor to calculate the heading.

The wind speed and direction are needed to calculate the track of the airplane.

We simply add the velocity vector with respect to the air to the wind velocity WRT to the ground and place this vector's origin at the coordinates of the plane WRT to the ground to get the track. .

flight track

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I feel like I am pointing the elephant in the room, but given only:

a) the position (coordinates from the ground) and

b) the magnitude of the velocity (aircraft's speed)

it is not possible to determine the flight path.

More information is required. A simple analogy is the following:

enter image description here

I am on number 2 and I move with a magnitude of velocity of 1 (absolute value), where will I be next?

If you don't know that you are moving towards positive or negative numbers, then you can be either at 3 or at 1.

2D example

In two dimensions the example would be the following: Your position is x=3 and y=4, and your velocity is 1 per time unit. Where will you be in the next time unit?

enter image description here

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