# Sum of the vectors from one fixed vertex to each remaining vertex of a regular polygon

I'm attempting to calculate the sum of the vectors from one fixed vertex of a regular m-sided polygon to each of the other vertices. It's for a study guide preceding my Linear Algebra exam tomorrow, and I'm entirely stumped by this question for some reason. The center of the polygon is at (0,0). How would I go about calculating this?

Thanks

Let the fixed vertex be at $(1,0)$, given by position vector $v_0$, and all the other vertices be given by position vectors $v_1$...$v_{m-1}$

Then $\Sigma_{i=0}^{m-1}v_i=0$, since they will form a closed polygon.

Now the required sum is $$(v_0-v_0)+(v_1-v_0)+(v_2-v_0)+...+(v_{m-1}-v_0)$$

$$=-mv_0=\left(\begin{matrix}-m\\0\end{matrix}\right)$$

• Oh, that absolutely makes sense, thanks! – Shrish Aug 30 '15 at 18:01
• Good luck with the exam! – David Quinn Aug 30 '15 at 18:03
• How do we know that they will form a closed polygon? – Scott Emmons Oct 3 '16 at 23:49
• @ScottEmmons since their sum is zero, they can be placed end to end to form a closed loop – David Quinn Oct 4 '16 at 5:34
• @DavidQuinn Which do we conclude first, that the sum of the vectors from the center to each of the vertices of the polygon is zero, or that the vectors from the center to each of the vertices of the polygon can be placed end to end to form a closed loop? I see how one is true if and only if the other is true, but I do not see at which we arrive first based on the given information of the problem. – Scott Emmons Oct 5 '16 at 14:54