Given two vectors u and v, their cross product u × v is a vector perpendicular to u and v. The motivation for this property, however, is never addressed. Here we show that the existence of the cross and dot products and the perpendicularity property follow from the concept of linear combination, which does not involve products of vectors. For our proof we consider the plane generated by a linear combination of u and v. When looking for the coefficients in the linear combination required to reach a desired point on the plane, the solution involves the existence of a normal vector n = u × v. Our results have a bearing on the history of vector analysis, as a product similar to the cross product but without the perpendicularity requirement existed at the same time. These competing products originate in the work of two major nineteen-century mathematicians, W. Hamilton and H. Grassmann. These historical aspects are discussed in some detail here. We also address certain aspects of the teaching of u × v to undergraduate students, which is known to carry some difficulties. This includes the algebraic and geometric denitions of u × v, the rule for the direction of u × v, and the pseudovectorial nature of u × v.