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Math Help - Quaternions and Rotation?

  1. #1
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    Quaternions and Rotation?

    I got this question in my Group Theory class and I'm not really sure how to go about it because I never really understood quaternions in the first place!

    Let v be the unit vector (  $\frac{1}{\sqrt{3}}$,$-\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$)

    and let R be the rotation through angle $60^\circ$ anticlockwise around Ov

    Calculate R(1,0,0) by conjugating a quaternion.

    Thanks for any help you can give me!

    P.S. That one in the first part of the vector should be on top, I couldn't get the MATH tags to wrok properly
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  2. #2
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    You are in my class, it appears. Unless our lecturer steals his questions from elsewhere.

    Section 23.4 in his notes cover this question pretty much perfectly.
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  3. #3
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    Oh well that's handy what section is relevant to question 3, do you know??
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  4. #4
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    I'm not sure if one is. I posted a thread here earlier looking for an answer, but nobody has responded yet.
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  5. #5
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    Quote Originally Posted by EuropeanSon View Post
    I'm not sure if one is. I posted a thread here earlier looking for an answer, but nobody has responded yet.
    If you don't show the forum you have tried anything, you won't get much help.
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  6. #6
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    The problem with that is that I have no idea where to start. I don't know how to calculate a unitary matrix from a quaternion (in actuality I have no idea how to form ANY matrix from a quaternion) and so without that basic knowledge can do no work. It's fairly elementary. I'm not asking for the answer to the question, just a piece of basic knowledge from which I can do some work.
    Last edited by Ackbeet; March 21st 2011 at 05:30 PM. Reason: Deleted objectionable material.
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  7. #7
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    Quote Originally Posted by Conn View Post
    I got this question in my Group Theory class and I'm not really sure how to go about it because I never really understood quaternions in the first place!

    Let v be the unit vector (  $\frac{1}{\sqrt{3}}$,$-\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$)

    and let R be the rotation through angle $60^\circ$ anticlockwise around Ov

    Calculate R(1,0,0) by conjugating a quaternion.

    Thanks for any help you can give me!

    P.S. That one in the first part of the vector should be on top, I couldn't get the MATH tags to wrok properly
    Every rotation of \mathbb{Re}^3, given by the axis u and the angle of rotation \alpha, is the result of conjugation by unit quaternions t=cos\frac{\alpha}{2}+\vec{u}sin\frac{\alpha}{2} (see here). Now you have \vec{u}, \alpha=-\frac{\pi}{3}, and v=(0, 1, 0, 0) in the link. Can you proceed from here?

    Note also that unit quarternions described by matrix (a+bi,c+di;-c+di,a-bi)=a1+bi+cj+dk, where 1=(1,0;0,1), i=(0, -1;1,0), j=(0, -i;-i,0), k=(i,0;0,-i) forms a group under multiplication which is isomorphic to SU(2) (see here).
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