Let's just talk about unit quaternions.(adsbygoogle = window.adsbygoogle || []).push({});

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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# B Any geometrical meaning of multiplication of quaternions?

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