# Show that the sum of vectors $p_i$ of a regular plane $n$-gon is the zero vector

This problem is from the book Abstract Algebra with applications by Spindler: Let $$P_1,\dots,P_n$$ be the vertices of a regular plane $$n$$-gon centered at $$O$$ and let $$p_i:=\overrightarrow{OP_i}.$$ Show that $$p_1+p_2+\cdots+p_n=0$$ i.e. that the sum of the vectors $$p_i$$ is the zero vector.

Solution: If $$n$$ is even, then it is trivial since each $$p_i$$ has an opposite vector, $$-p_i$$. There are $$n/2$$ of both sets of vectors, and thus their sum is zero.

So, suppose $$n$$ is odd and consider when $$n=5.$$ Observe that the opposite of each $$p_i$$ bisects the side of two adjacent vectors e.g. $$-\lambda p_1$$ bisects side formed by $$p_3$$ and $$p_4$$ where $$0<\lambda\leq1\enspace[\color{red}1]$$

We then have, $$-\lambda p_1=\frac{1}{2}(p_3+p_4)$$ $$-2\lambda p_1=(p_3+p_4)\enspace[\color{red}2]$$

Questions:

$$[\color{red}1]$$ What is $$\lambda$$ suppose to be? I know for this case, $$n=5$$, it has to be less than 1 but when $$n=3,\:\lambda=\frac{1}{2}$$. But for the triangle its easy to see that $$p_1+p_2+p_3=0$$ since $$p_2+p_3=-p_1$$ as can be seen below.

$$[\color{red}2]$$ I'm not sure what to do after this sense I need to know what $$\lambda$$ is.

The following is to help visualize the problem:

• This fact does not require such complicated argument. $p_1 + p_2 + ... + p_n =0$ because when you rotate all the vectors by $\frac{2\pi}{n}$ their sum on the one hand does not change and on the other hand it also rotates by $\frac{2\pi}{n}$, so it must be zero Apr 2, 2019 at 20:04
• This is a duplicate of MSE question 2276723 "Property of Vectors of an $n$-gon". Apr 2, 2019 at 20:35
• It is not an exact duplicate, as this question is about a specific proof of this property. Apr 2, 2019 at 21:02

The exact value of $$\lambda$$ is not relevant, it is instead necessary that $$\lambda$$ be the same for all $$n$$ vertices. If you add up the $$n$$ equations you get: $$-2\lambda(p_1+p_2+\cdots+p_n)=2(p_1+p_2+\cdots+p_n), \quad\text{hence:}\quad p_1+p_2+\cdots+p_n=0.$$
• You can rewrite the first equation as $$2(1+\lambda)(p_1+p_2+\cdots+p_n)=0$$. Apr 2, 2019 at 20:58