Answer
See explanations.
Work Step by Step
Step 1, Assume the coordinates for the end points of vectors are $(x_1,y_1), (x_2,y_2), (x_3,y_3), ... (x_n,y_n)$, the vectors can be written as $\vec{v_1}=\langle x_2-x_1, y_2-y_1 \rangle$, $\vec{v_2}=\langle x_3-x_2, y_3-y_2 \rangle$, ... $\vec{v_n}=\langle x_1-x_n, y_1-y_n \rangle$.
Step 2. The sum the all the vectors can be written as $\vec{v_s}=\langle \sum (x_k-x_{k-1}), \sum (y_k-y_{k-1}) \rangle$ where $k=1, 2, 3, ... n$ and use $n$ for $k-1$ when $k=1$
Step 3. As all the vectors form a closed polygon, the results of the sums are zeros which gives $\vec{v_s}=\langle 0, 0 \rangle$
