The origin is at the center of a regular m-sided polygon. what is the sum of the vectors from one fixed vertex to each of the remaining vertices

-m * (vector from center to the fixed vertex)

Step-by-step explanation:

The sum of concern is the sum of difference vectors (vn-vf), where vn is the vector from center to vertex n, and vf is the fixed vertex.

This sum can be divided into two parts:

... ∑[n=1 ... m] (vn - vf) = ∑[n=1 ... m]vn - m (vf)

Sine the polygon is regular the sum of vectors from its center to each vertex is zero, so this sum reduces to ...

... 0 - m (vf)

... = - m * (vector from center to fixed vertex)