(adsbygoogle = window.adsbygoogle || []).push({}); Is there a "vector" version of The Elements?

I have noticed that many theorems in classical geometry can be proven using vectors. Thus, I am naturally inclined to believe that pretty much every theorem in the Elements can be proven using vectors (using the vector space axioms and the vector definitions of norm, angle, and parallel, rather than the axioms, postulates and definitions adopted by Euclid).

First of all, is this possible? What would be the vector form of Euclid's 5th (parallel) postulate be? After all, I know that Euclid's 5th postulate is the basis of Euclidean geometry, so certainly a construction of the Elements that uses vectors must have some vector analog of the 5th postulate, otherwise it would produce non-Euclidean geometries?

If it is possible, wouldn't it be circular? Don't some of the vector space axioms follow from Euclidean geometry? Or are the vector space axioms chosen to be consistent with the axioms and theorems of Euclidean geometry, in which case it does not really matter?

And finally, if it is possible, has it been done? What's the name of the text in that case? I am particularly interested in the usage of vectors to prove theorems involving circles and tangents from Euclidean geometry.

I am only speculating, so I deeply apologize if my idea sounds dumb. In such case I would appreciate an explanation of why this (construction) is not possible.

Thanks!

BiP

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# Is there a vector version of The Elements?

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