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You speak of a multiplication operation that "works", also stating that it's not always possible for a finite dimensional real vector space. Doesn't the definition of a real vector space require multiplication that "works" (through the scalar multiplication requirement) or are you referring to the multiplication between vectors within that space "working"?

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Multiplication of vectors. That’s the meaning of the algebra word in the phrase, finite dimensional associative division algebra. In R^3 you can’t define multiplication of vectors in a way that works. That is, in a way that each vector can be multiplied by itself and in which each vector has a multiplicative inverse while preserving the real vector space structure.

Thanks! I'm still in the process of really grokking linear algebra as I only just took a class on it, so this is helpful.

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