For (x,y) in R2, describe the set of orbits geometrically.

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In summary, the proof states that if x and y are real numbers, then the set (R,+) acts on R2 by adding r to the first coordinate of a point (x,y). For part (b) of the question, the orbit of a point (x,y) in R2 is the set of all points in R2, and geometrically, it represents a circle centered at the point (x,y). Points that are in the same orbit have the same first coordinate, while points in different orbits have different first coordinates.
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Edellaine
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Homework Statement


4a) If x, y are in R, prove that (R, +) acts on R2 by (x,y)*r = (x+r, y) for all (x,y) in R2 and for all r in R.
b) If (x,y) are in R2, find the orbit of (x,y). Describe geometrically.


Homework Equations


none that I can think of


The Attempt at a Solution



The group action part is easy. I have problems with (b). Am I supposed to just give the definition of the set (I don't think so).

The set of orbits, geometrically, seems to be the set of all points (x,y) in R2.
 
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  • #2
The question is to describe single orbits geometrically (1,1) and (2,1) are in the same orbit, right? (1,2) isn't in the same orbit as the other two. Describe the difference geometrically.
 

1. What does "orbits" mean in this context?

In mathematics, an orbit is a set of points that are related by a certain transformation or action. In this case, the points are in the Cartesian plane (R2) and the transformation is described by the variables x and y.

2. How do you describe a set of orbits geometrically?

A set of orbits in R2 can be described geometrically using a graph or diagram. The points in each orbit will follow a specific pattern or path, depending on the transformation defined by the variables x and y. These paths can be represented by curves, lines, or other shapes on the graph.

3. What does the notation "For (x,y) in R2" mean?

This notation means that the set of points being described is in the Cartesian plane, which is represented by R2. The variables x and y represent the coordinates of each point in the plane.

4. How many orbits are there in a set of points in R2?

The number of orbits in a set of points in R2 will depend on the specific transformation or action being applied. In some cases, there may be only one orbit, while in others there may be multiple orbits.

5. Can you give an example of a set of orbits in R2?

One example of a set of orbits in R2 is the set of points that form a circle when plotted on a graph. In this case, the transformation is a rotation around a fixed point, and the orbits are the points that follow the circular path. Another example could be a set of points that form a line when plotted, where the transformation is a translation along that line.

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