I'm having a hard time trying to phrase the specific question, but the simplest one I've found is to point to This link here.

Like the page says, laws have been making mathmatical foundations quite intuitive, which I think have been obscuring the core details within. To be specific towards this question, I'd like to know what were the grounds to proving vector addition?
After searching numerous articles on the definition of vector addition, I finally found a little twisted definition in the "vector calculus" book by M. Coral, which states that A vector + B vector means simply translating A vector's tail point to B vector's head point. But, the confusion is, how was this first founded, this mechanism developed, therefore, the grounds to proving this. If we take the aid of something like the triangle law, or the parallelogram law, it would be a biased proof, would not really make sense to try to prove something from the base itself.

I'm not one with a vast knowledge to dive into something like Newton's book of principia mathmatica, which the aforementioned link says has a bit of background, so I'm asking fellow wise stackexchangers if there's something to this?

• I think this question should be posted on physics stackexchange. In mathematics, it's just so defined. – velut luna Dec 18 '16 at 14:59
• The source is "physical" : Newton and before: addition of forces but also "addition" of movements. – Mauro ALLEGRANZA Dec 18 '16 at 15:01

The first occurrence is presumably :

Now whenever a body is moved in two directions in a fixed ratio it necessarily travels in a straight line, which is the diagonal of the figure which the lines arranged in this ratio describe.

See also : Sanilo Capecchi, Historical roots of the rule of composition of forces (2012).