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

[Edellaine]

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.

 

[Dick]

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.