Vector Addition is Associative/Proof 1

Proof

 * Vector-Addition-is-Associative.png

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be positioned in space so they are end to end as in the above diagram.

Let $\mathbf v$ be a vector representing the closing side of the polygon whose other $3$ sides are represented by $\mathbf a$, $\mathbf b$ and $\mathbf c$.

By the Parallelogram Law we can add any pair of vectors, and add a third vector to their resultant.

Hence we have: