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I noticed that I can get a quaternion by adding the scalar value of a dot product between two vectors to the cross product of the same two vectors. Is that just pure coincidence, or is that how it was "meant" to be?

Some loony with a calculator
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  • Can you give examples? I think thus is right but need some examples to analyze. – Oscar Lanzi Jun 26 '20 at 20:30
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    This isn't quite accurate; the scalar term in a product of two quaternions is the product of the scalar terms minus the dot product of the 'imaginary' portions of the two quaternions. But no, it's certainly not coincidence. You might want to see https://math.stackexchange.com/questions/984438/is-there-a-relationship-between-the-cross-product-and-quaternion-multiplication?rq=1 and https://math.stackexchange.com/questions/752070/motivation-for-construction-of-cross-product-quaternions?noredirect=1&lq=1 and a couple of other questions on this site... – Steven Stadnicki Jun 26 '20 at 20:31
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    By "get a quaternion", perhaps you mean the scalar-vector split multiplication$$(p_s+\vec{p}_v)(q_s+\vec{q}_v)=(p_sq_s-\vec{p}_v\cdot\vec{q}_v)+(p_s\vec{q}_v+q_s\vec{p}_v+\vec{p}_v\times\vec{q}_v).$$(I've edited my comment to use Wikipedia's notation, so you can find their discussion more easily. They don't put arrows over the vectors. @Andrei's reference below is to the original version of my comment, which had $p_s=a,,q_s=b$.) – J.G. Jun 26 '20 at 20:33
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    In @J.G. 's comment, if $a=b=0$, you get a scalar product and a vector product – Andrei Jun 26 '20 at 20:35

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