
operation on sphere
Hi;
Let S^2 be a 2dimensional 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.
Thank you in advance

Re: operation on sphere
By rotating a vector along v do you mean rotating it around v, i.e., this?
https://lh5.googleusercontent.com/l.../rotation1.png
If so, I don't see how the result of rotation of anything other than v can coincide with v...

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

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.