Vectors and Quaternions

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I WOULD like to ask Prof. Knott whether there would be any fatal objection to defining the scalar product of two vectors as equal to the product of their tensors into the cosine of the angle between them, so that, if the vectors are , and , the scalar product would be , and not . If this is done, and, for the sake of associativeness of products, i2 is made equal to - 1, the distributive or quaternionic product of two (or more) vectors would be their vector product minus their scalar product. The change suggested would enable students to gradually accustom themselves to the notation of the calculus, which would in fact then form an abridged notation for the cartesian expressions and operations which enter into physical investigations.

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LODGE, A. Vectors and Quaternions. Nature 48, 198–199 (1893) doi:10.1038/048198c0

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