1. ## operation on sphere

Hi;

Let S^2 be a 2-dimensional unit sphere in R^3. Let u and v be two unit vectors in R^3 (so in S^2). We will define an operation on S^2 by u*v is a unit vector obtained by rotating a vector u along v by 180 degree (Pi).

Define stab(v)={u in S^2: u*v=v}. My question is find stab(v) where v is in S^2.

2. ## Re: operation on sphere

By rotating a vector along v do you mean rotating it around v, i.e., this?

If so, I don't see how the result of rotation of anything other than v can coincide with v...

3. ## Re: operation on sphere

Thank you Mr. Emakarov for your answer. If v is the rotation axis, then I also think like you u*v=v only if u=v. But may be I missunderstood the definition because our instructor said no the answer is not correct. I will ask him to interpret what the binary operation is ?

thank you so much

4. ## Re: operation on sphere

I checked the definition is correct. I got the answer, in fact stab(v)={v,-v}, v and -v are antipodal and distinct!

Thank you all for your contribution.