The Eight Vertices of a Cube: A Connection Between Coordinate Geometry, Vector Algebra and The Algebraic Structure Z₂³

This paper investigates the correspondence between the eight vertices of a unit cube, their coordinate representations in three-dimensional Euclidean space, their associated vectors, and the finite algebraic structure Z₂ × Z₂ × Z₂. Each vertex has coordinates whose entries belong to {0,1}. The standard basis vectors i, j, and k represent the coordinate axes, while the eight binary triples form a vector space of dimension three over the field Z₂. The paper distinguishes ordinary vector addition over the real numbers from componentwise addition modulo 2, establishes basic algebraic properties, and discusses applications to digital electronics, coding theory, graph theory, and finite algebra.