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Math Help - operation on sphere

  1. #1
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    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.

    Thank you in advance
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  2. #2
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    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...
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  3. #3
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    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
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  4. #4
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    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.
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