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Why dot product of A and B is A.B=ABcos(theta)?

Could it be that dot product did not have any physical meaning initially, but scientists would research on many physical quantities and somehow found a correlation between vectors associated with the quantity such that product of the two vectors associated and cosine of the angle between them would always yield a scalar quantity which is equal to the magnitude of the quantity they are dealing with? Or were there something different notion which could prove two vectors would yield a scalar quantity when they are multiplied the way I mentioned above?

Same question for cross product as well.

Please enlighten me with knowledge required to grasp this thing otherwise I might not be able to understand such concepts

MSKB
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  • Isn't this more suitable for math.SE or even history of science.SE? (Heaviside and Gibbs introduced the products as a simplification of Hamilton's quaternionic algebra - the products came from mathematical considerations, although interest in this particular math was largely due to the need for vector quantities in electromagnetism.) – Anders Sandberg Jun 22 '21 at 14:50
  • I literally forgot about the history S.E. Thanks a lot – MSKB Jun 22 '21 at 14:53
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    As a hint, another name for the dot product is the projection product, and the dot product of an arbitrary vector and a unit vector gives the magnitude of the projection of the first vector onto the direction of the unit vector. –  Jun 22 '21 at 14:56
  • Does this answer your question? why do we use cosine as the expression of vector dot product? – Sandejo Jun 22 '21 at 16:23
  • If you multiply a vector with another vector with reference to another vector - the dot product will be the real part (the component in the direction of the other vector) and the cross product will be in direction perpendicular to it. – Moti Jun 22 '21 at 16:24
  • @Moti this mathematical interpretation we know is because of the discovery of this equation. But how did the one who discovered this come into a conclusion that this is the only equation which would satisfy his query? – MSKB Jun 22 '21 at 17:14
  • You are asking about the meaning of dot product - it is not easy to "uncover" the ahah moment. The physical interpretation/meaning is the component of a vector in the direction of another vector - as example determining "closing velocity". If you have two moving object with a certain distance - This will be the ONLY component to determine the time for collision. – Moti Jun 22 '21 at 17:41

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