The origin is at the center of a regular m -sided polygon. a) What is the sum of
ID: 2960808 • Letter: T
Question
The origin is at the center of a regular m-sided polygon.
a) What is the sum of the vectors from the origin to each of the vertices of the polygon? Give your reasoning.
My answer:
b) What is the sum of the vectors from one fixed vertex to eaech of the remaining vertices? (Hint: You should use an algebraic approach along with your answer to part a)
I need help on part b.
Assume the circumradius of the polygon is 1. Project the polygon onto the complex plane, allowing each vertex of the m-sided polygon to become one of the mth roots of unity. Each vertex can therefore be expressed as omega k for all {k |k Z 0 k m - 1}, with omega = e 2pi i/m Thus the sum of the vectors from the origin to each vertex is the sum of the roots of unity omega0, omega1, . . . , omega m-1 = 1 - omega m/1 - omega by the sum of a geometric series. The m th root of unity is 1 and therefore the sum of the roots of unity is 0, implying that the real components and the imaginary components both independently converged to 0. This inference also implies that the sum of the y components of the vectors and the sum of the y components of the vectors are both zero as well, and thus the sum of all of the vectors is therefore the zero vector. It is therefore valid to assume the circumradius is 1 because the sum shall continue to be the zero vector regardless of the actual circumradius.Explanation / Answer
ASSUMING THAT IS A REGULAR POLYGON WITHH ALL SIDES AND ANGLES SAME WITH CIRCUM RADIUS = 1 UNIT . LET US TAKE OA1 AS THE INITIAL LINE AND TAKE ALL ANGLES WRT THAT .. SINCE THE POLYGON IS REGULAR , EACH CENTRAL ANGLE A1OA2