Vector geometry.Geometric Proof for the sum of vectors.

Question

Why the sum of 2 vectors is the diagonal of the Parallelogram they create.Proof.Is that the definition or is there a proof .Using the Parallelogram law say. x and y be the sides of the parallelogram and $a,b$ the 2 diagonals from the parallelogram law i know $2x^2+2y^2=a^2+b^2$ .If i assume $a=x+y$ and $b=x-y$ i know the law will be verified but that doesn't mean the diagonal $a$ is the one i want it to be the sum i can change $a$ to be $b$ and geometrically the vector supposed to be that sum of $x+y$ will be the $x-y$. Also i could verify the law with many other values for $a$ and $b$ so why the sum of the vectors IS the diagonal ?

Answer 1

It's more a side effect: on the plane defined by two vectors $\vec{V1}$ and $\vec{V2}$, according to Chasles, it is commutative and the sum $\vec{V} = \vec{V1} + \vec{V2} = \vec{V2} + \vec{V1}$.

It implies both triangles $(\vec{V1},\vec{V2},-\vec{V})$ and $(\vec{V2},\vec{V1},-\vec{V})$ form a parallelogram $(\vec{V1},\vec{V2},-\vec{V1},-\vec{V2})$ that has $\vec{V}$ for diagonal.