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I wanted to ask about how they arrived to the rule of addition of vectors. How did they know that if we add the X's and Y's of two vectors they would get a third vector which has exactly the same direction and magnitude of the force that could replace these two vectors or forces.

I'm convinced that it's correct and can feel or see the direction a point will accelerate in if two certain forces are applied to it. And can feel how the resultant force tends to get closer to the bigger force and how if two equal forces are applied with the same angle the resultant force is going to be exactly between them.

So I'm sure who invented the vectors had the same feelings and visions too but how did he arrive at this simple method to get such fascinating and exact results, not only did he manage to get the direction but the magnitude of the resultant too!

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  • $\begingroup$ Good questions, but it is not "we arrive" but "they arrive". Take a look here and here for some references. $\endgroup$
    – Moishe Kohan
    Commented Jan 24, 2021 at 18:21
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    $\begingroup$ The first known occurrence is in Pseudo-Aristotle's Mecanicha. See Miller, The Parallelogram Rule from Pseudo-Aristotle to Newton(2017): the "proof" is geometrical. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Jan 24, 2021 at 18:30
  • $\begingroup$ @Moishe Kohan I used "they" but then my question was edited into "we" ;( and thanks for the reference :D $\endgroup$
    – Manar
    Commented Jan 24, 2021 at 18:44
  • $\begingroup$ Your question might be more useful if you asked "Who first formulated the rule... and how did they prove it?" $\endgroup$
    – Carl Witthoft
    Commented Jan 25, 2021 at 12:30
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    $\begingroup$ People had these "feelings" long before invention of the vectors in late 19th century. Something like the parallelogram of velocities is used by Archimedes in On Spirals to find tangents, Newton used it for both velocities and forces in Principia. By the time vectors came to be used by Gibbs to organize mechanics there was nothing left to arrive at. $\endgroup$
    – Conifold
    Commented Jan 25, 2021 at 21:20

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