The operations of two vector multiplication (the scalar and vector products) are introduced in physics and mathematics textbooks just as a definition, without any reference or discussion on the formal reasons and/or motivations that have led to these structures. In this work, a short pedagogical review on the origins of these definitions is presented. We discuss the formal results obtained by Hamilton in the context of quaternionic algebra and some "changes" performed by Gibbs and Heaviside, leading to what is now usually known as "vector algebra". We present comments on some disadvantages of these "changes", referring to more practical and formal systems (Grasmmann and Clifford algebras). Some basic and recent works on the subject are also mentioned and commented.
vector algebra; quaternions
