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.
Best Answer
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 $$(v_0-v_0)+(v_1-v_0)+(v_2-v_0)+...+(v_{m-1}-v_0)$$
$$=-mv_0=\left(\begin{matrix}-m\\0\end{matrix}\right)$$