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Regarding user Muphrid's answer to the posted question why do we use cosine as the expression of vector dot product?, would it be possible to pictorially express this thought/answer? It would be extremely useful, as I do not know how to determine the rotation in a given plane, therefore I cannot prove to myself that cosine is required in the dot product definition.

Thanks in advance. And, Thanks question OP Shams Tarek, Thanks Muphrid.

Gonçalo
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Good Day
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  • Is here a reason you chose that specific answer? The other answers more clearly depict why the cosine shows up in the results. – Triatticus Jan 07 '25 at 00:47
  • It is true, the other answers do provide a good explanation. Regarding Muphrid's answer, it stated that cosine is involved for the dot product definition when the vectors are in the Euclidean plane, and so I was wondering how can the plane at hand be determined. Also, why is the dot product defined by hyperbolic cosine, given that the Lorentzian plane seems to be attained from the Euclidean plane by rotating the Euclidean by (3/4)*pi in the clock-wise direction? – Good Day Jan 07 '25 at 16:54
  • No the lorentzian part is a completely different object altogether, in that case one of your dimensions is negative relative to the others in the metric that describes it. – Triatticus Jan 07 '25 at 17:12
  • Got it, thanks for this input. – Good Day Jan 07 '25 at 21:31

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