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B Any geometrical meaning of multiplication of quaternions?

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  1. Dec 12, 2017 #1
    Let's just talk about unit quaternions.
    I know that $$\left(\cos{\frac{\theta}{2}}+v\sin{\frac{\theta}{2}}\right)\cdot p \cdot \left(\cos{\frac{\theta}{2}}-v\sin{\frac{\theta}{2}}\right)$$
    where ##p## and ##v## are purely imaginary quaternions, gives another purely imaginary quaternion which corresponds to ##p## rotated by an angle ##\theta## about the axis specified by ##v##. So the product ##q\cdot p \cdot q'## has a geometrical meaning.

    But what about any arbitrary unit quaternion multiplication ##q_1\cdot q_2##? What does it mean geometrically (just like unit complex number multiplication means adding their angles)?

    If ##z_1\cdot z_2=z_3##, then ##z_3## is the point we end up at when we rotate point ##z_1## by the argument of ##z_2## or vice-versa. Now, if ##q_1\cdot q_2=q_3##, then ##q_1,q_2,q_3## are points in four dimensions. What is the relation between these three points?
     
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  3. Dec 12, 2017 #2

    FactChecker

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    It gets complicated. In my opinion, the best geometric treatment of 3D and higher is with Geometric Algebra. It systematically treats many geometric concepts with algebraic operations.
    (see )

    PS. A word of warning. Although Geometric Algebra consolidates and replaces a lot of specialized mathematical "gimic" algebras, it is not well known, the learning curve is not trivial, and translating the more popular algebras to it is not always easy.
     
  4. Dec 12, 2017 #3

    fresh_42

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    For the unit quaternions we have ##U(1,\mathbb{H}) \cong SU(2,\mathbb{C}) \cong \mathbb{S}^3## (cp. section 3 in https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/).
    Thus the multiplication is the same as in the special unitary group, or on the 3-sphere. Both, ## SU(2,\mathbb{C}) ## and ## \mathbb{S}^3 ## are geometric objects, so the multiplication directly translates into geometry.
     
  5. Dec 12, 2017 #4

    StoneTemplePython

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    I tried a book on geometric algebra a while ago but it didn't speak that much to me.

    For quarternions, thinking about them as part of special unitary group, and in particular using a matrix representation, as done in Stillwell's Naive Lie Theory, is how I'd do it.
     
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