Unique point determined by 3 distances

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In summary, the conversation discusses the concept of determining a point in the complex plane by its distances from three given complex numbers, and presents a proof by contradiction to demonstrate its uniqueness.
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Homework Statement


Suppose u,v,w are three complex numbers not all on the same line. Prove that any point in z in the complex plane is uniquely determined by its distances from these 3 points.


The Attempt at a Solution


I can explain this by saying something like, if u,v,w are points and given different radii for each of them, their circles will overlap and then one point is created. It's very messy, but I don't really know how to explain it. I also would like to show some sort of computation to prove it as well but can't seem to get my head around it.
 
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I haven't done this, but here is an approach, using a proof by contradiction. Instead of proving that any point in z in the complex plane is uniquely determined by its distances from u, v, and w, suppose that there are two points, z1 and z2 that are the same distances from u, v, and w. IOW, |z1 - u| = |z2 - u|, and so on for the distances to v and w.

If you arrive at a contradiction, that will prove what you really want to prove.
 

1. What is a "Unique point determined by 3 distances?"

A unique point determined by 3 distances is a mathematical concept that describes the location of a point in space, given its distance from three known points. It is also known as the trilateration problem.

2. How is a unique point determined by 3 distances calculated?

To calculate a unique point determined by 3 distances, the three known points and their corresponding distances must be used to form three circles. The intersection point of these circles is the unique point. This can be done using the Pythagorean theorem and algebraic equations.

3. What are the applications of unique point determined by 3 distances?

This concept has various applications in fields such as geolocation, navigation, surveying, and robotics. It can be used to pinpoint the location of an object or to track the movement of an object in a given space.

4. Can a unique point determined by 3 distances be calculated in 2D and 3D space?

Yes, this concept can be applied in both 2D and 3D space. In 2D space, the three distances determine the location of the point on a 2D plane. In 3D space, the three distances determine the location of the point in 3D coordinates.

5. What are the limitations of unique point determined by 3 distances?

One of the limitations of this concept is that it requires accurate measurements of the distances and assumes that the known points are fixed. Any errors or uncertainties in the measurements can lead to incorrect results. Additionally, this method cannot be used to determine the location of a point if it lies on the same line as two of the known points.

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