# operation on sphere

• Dec 25th 2013, 03:36 AM
student2011
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.

• Dec 25th 2013, 12:51 PM
emakarov
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...
• Dec 25th 2013, 08:32 PM
student2011
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
• Dec 26th 2013, 10:08 AM
student2011
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.