Revisiting the Definition of Vectors—From ‘Magnitude and Direction’ to Abstract Tuples

Vectors are almost always introduced as objects having magnitude and direction. Following that idea, textbooks and courses introduce the concept of a vector norm and the angle between two vectors. While this is correct and useful for vectors in two- or three-dimensional Euclidean space, these concepts make no sense for more general vectors, that are defined in abstract, non-metric vector spaces. This is even the case when an inner product exists. Here, we analyze how several textbooks are imprecise in presenting the restricted validity of the expressions for the norm and the angle. We also study one concrete example, the so-called ‘vector-based sustainability analytics’, in which scientists have gone astray by mistaking an abstract vector for a Euclidean vector. We recommend that future textbook authors introduce the distinction between vectors that have and that do not have magnitude and direction, even in cases where an inner product exists.