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How can we define something like the dot product $u\cdot v = |u||v|\cdot \cos[u,v]$ and then use it to find the angle between to vectors? Mustn't there necessarily be some truth to the definition? Couldn't we define the dot product in some other way and then obtain the some other false angle between the vectors.

i.e why is the dot product definition the right definition?

Davide Giraudo
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user370043
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    The Theorem is that this definition coincides with the other. that is, $\left(a,b\right)\cdot \left(c,d\right)=ac+bd$. There is enormous content in that result. – lulu Sep 18 '16 at 13:02
  • Here's a proof of the equivalence between the algebraic form of the dot product, and the geometric one you mention above. There is truth there: http://math.stackexchange.com/questions/509719/proof-of-equivalence-of-algebraic-and-geometric-dot-product – KR136 Sep 18 '16 at 13:21

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