# Equivalence relation (geometry)

1. Jan 20, 2014

### Lee33

1. The problem statement, all variables and given/known data

Let $\mathbb{R}^2 = \{Q = (a,b) | a,b\in \mathbb{R}\}$. Prove that if $q_1 = (a_1,b_1)$ and $q_2=(a_2,b_2)$ are equivalent, meaning $a_1^2+b_1^2 = a_2^2 +b_2^2$, then this gives an equivalence relation on $\mathbb{R}^2$. What is $[(1,0)], [(0,1)],[(2,2)],[(0,0)]?$ What does an equivalence class look like?

2. The attempt at a solution

I know how to do the first part with the equivalence relation but I am not sure how to do the second part of the question?

2. Jan 20, 2014

### Dick

I'm not sure what you are asking about. I assume [(0,1)] means the equivalence class of (0,1). What does that look like?

3. Jan 20, 2014

### Lee33

Yes, it is the equivalence class but I don't understand how it looks like. Is it just a unit circle?

4. Jan 20, 2014

### Dick

Well, yes. If (x,y) is related to (0,1) then x^2+y^2=0^2+1^2=1. That's the equation of the unit circle.

5. Jan 20, 2014

### Lee33

Gotcha and that goes for (1,0) which will be a circle centered at 0 with radius 1 and how about (2,2)?

6. Jan 20, 2014

### Dick

It's pretty similar to the other one, isn't it? You tell me what kind of circle it is.

7. Jan 21, 2014

### Lee33

I understand now. Thanks for the help!