Above, we draw a parallelogram with sides constructed by vectors $u$ and $v$ (and their parallels). Rotating the graph, you can view the shape in three dimensions, with the sum $w = u + v$ drawing the diagonal (and $u-v$ drawing the other one). This is analogous to the two-dimensional picture, just embedded in 3-D, showing this image exists in higher dimensions.
\begin{remark}
This geometric picture has a clean algebraic counterpart, written entirely in terms of norms. Herein, we prove a geometric fact completely algebraically. The diagonals of the parallelogram are $u+v$ and $u-v$, and adding their squared lengths gives the \textbf{parallelogram law}
$$\|u+v\|^2 + \|u-v\|^2 = 2\|u\|^2 + 2\|v\|^2.$$
This holds in $\R^n$ for any $n$, and the cross terms $\langle u,v\rangle$ and $-\langle u,v\rangle$ appearing when you expand $\|u+v\|^2$ and $\|u-v\|^2$ cancel exactly.
\end{remark}