On the sum of the angles between three vectors

TL;DR

This paper provides a necessary and sufficient condition for the sum of the angles between three vectors in 2 to be 244, generalizing Euclid's theorem on the interior angles of triangles.

Contribution

It introduces a new condition characterizing when three vectors' pairwise angles sum to 24, extending Euclid's theorem to a vector angle context.

Findings

Characterization of when three vectors' angles sum to 24

Proof of Euclid's theorem as a special case

Reduction to angle multiples of 4 and betweenness relations

Abstract

For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2\pi$. As an easy consequence of this, a proof of Euclid's theorem that the sum of the interior angles of any triangle is $\pi$ is provided. So, the main result of this note can be considered a generalization of Euclid's theorem. To a large extent, the consideration is reduced almost immediately to a choice for the sum of three related angles among the three integer multiples $0,2\pi,4\pi$ of $\pi$. The rest of the consideration concerns only various betweenness relations.