The following parallelogram has one corner at the origin. The two neighboring corners are given by vectors $\mathbf{a}$ and $\mathbf{b}$. Express the fourth corner as a vector.
If we move the tail of the vector $\mathbf{a}$ to start at the point $\mathbf{b}$, we end up at the fourth corner.
The vector from the origin to that fourth corner is thus $\mathbf{a} + \mathbf{b}$.