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

Question

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. The center of the polygon is at (0,0).

It's for a study guide preceding my Linear Algebra exam tomorrow, and I'm entirely stumped by this question.

Answer 1

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 $$(v_0-v_0)+(v_1-v_0)+(v_2-v_0)+...+(v_{m-1}-v_0)$$

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

Answer 2

Let $V_1,V_2,\ldots,V_m$ be the regular polygon's vertices and $O$ its centre.

Then the sum of the vectors from vertex $V_m$ to each of the other vertices is $$\vec{V_mV_1}+\vec{V_mV_2}+\ldots+\vec{V_mV_{m-1}}\\= (m-1)\vec{V_mO}+(\vec{OV_1}+\vec{OV_2}+\ldots+\vec{OV_{m-1}})+(\vec{OV_m}-\vec{OV_m})\\= m\vec{V_mO}+(\vec{OV_1}+\vec{OV_2}+\ldots+\vec{OV_m})\\= m\vec{V_mO},$$ since, by symmetry, $(\vec{OV_1}+\vec{OV_2}+\ldots+\vec{OV_m})=\mathbf0.$