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Before, I elaborate my question, I would mention that this question is similar to many questions asked on this site. Still, none of the answers satisfied me.

Addition and substraction of vectors seems simple enough. My physics teacher told me this:-

Attach two strings to an object and pull it from different directions at once with different forces. The object does not move towards only one of the forces, but somewhere towards the middle (depends). A genius guy observed this phenomenon and this led to the triangle/parallelogram law of vector addition. This explanation seems simple enough.

Now, the confusing part. The multiplication of vectors is defined mathematically. But, this math came from some observations, did'nt it? What were the the observations that led to the use of scalar and vector product in physics?

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    $\begingroup$ Dot product: e.g. work done. Cross product: for instance, Lorentz force law. $\endgroup$
    – JamieBondi
    Commented May 29, 2017 at 6:52
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    $\begingroup$ I'd also like to partially debunk the notion that "math comes from observations." Though in many cases, physical observations drive the development of new mathematical tools, it is also possible that physical insight can come from existing mathematics. If string theory ever gets off the ground, it would be a great example of this. $\endgroup$
    – probably_someone
    Commented May 29, 2017 at 7:18
  • $\begingroup$ There are many answers to similar questions on this site - as you seem to be aware - so I suspect you want an historically accurate account. This question might therefore fare better on History of Science and Mathematics Stack Exchange. I believe it was Gibbs and Heaviside who popularized if not defined these products and I believe they drew on Hamilton's work on quaternions. The dot and cross product are the real and imaginary parts of the quaternion product (see here) and quaternions represent rotations. So, e.g. the cross product is .... $\endgroup$
    – Selene Routley
    Commented May 29, 2017 at 7:27
  • $\begingroup$ ... the Lie bracket in the Lie algebra of the unit quaternion group (what Hamilton called "versors") and thus represents an infinitessimal rotation about an axis: $\vec{\omega}\times \vec{r}$ is the time derivative of the point at position $\vec{r}$ when Euclidean space is rotated at angular velocity $\vec{\omega}$ about the origin. $\endgroup$
    – Selene Routley
    Commented May 29, 2017 at 7:30
  • $\begingroup$ Thomas Pynchon perhaps said it best in Against The Day: "Anarchists always lose out, while the Gibbs-Heaviside Bolsheviks, their eyes ever upon the long-term, grimly pursued their aims, protected inside their belief that they are the inevitable future, the xyz people, the party of a single Established Coordinate System, present everywhere in the Universe, governing absolutely. We were only the ijk lot, drivers who set up their working tents for as long as the problem might demand, then struck up camp again and moved on, always ad hoc and local, what do you expect?" $\endgroup$
    – David Hammen
    Commented May 29, 2017 at 8:18

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