# Math Help - sum of radius vectors

1. ## sum of radius vectors

A circle with center at O is divided into n equal arcs, n≥ 2, by the
points A₁A₂ A₃....A_{n} Find the sum of radius vectors i OA_{i}
where
, i=1,2,...,n.

2. Let's use the circle whose radius is $1$ in the complex plane. Assume that $A_{0}=1$.

Your points are the $n^{th}$ roots of the unity. Let $\omega$ be a $n^{th}$ primitive root of the unity (i.e. $\omega^{n}=1$ and $\forall d).
Then your sum is $1+\omega +...+\omega^{n-1}=0$

If you want to generalize this to your problem, just apply a homothety and a rotation to the circle we used to obtain yours.

3. Hello, makenqau!

A circle with center at $O$ is divided into $n$ equal arcs, $n \geq 2$
by the points: $A_1, A_2, A_3, \hdots A_n.$
Find the sum of radius vectors: . $\sum^n_{i=1} \overrightarrow{OA}_i$

This is a classic problem . . .

I recall that the sum is zero, but the proof eludes me.

With that in mind, you may be able to construct your own proof.