Cross product and scalar triple product

British mathematician William Hamilton's wild dream was to discover a number system capable of turning the various isometries of space into simple additions or multiplications. His discovery of the quaternions in 1843 strengthened this ambition. A quaternion is a pair consisting of a scalar and a vector. Multiplying (0, u) by (0, v) yields two components: the dot product (up to sign) and the cross product, both unknown at the time. The physicist Josiah Gibbs immediately grasped the value of these two concepts. Much to Hamilton's dismay, he preferred to define them directly, without using quaternions. He later discovered that they already appeared in Grassmann's work, published in 1844. Other scientists, such as Oliver Heaviside, followed his lead and pursued this research, introducing concepts associated with the study of scalar- or vector-valued functions defined on a vector space. This led to the definition of the gradient, divergence and curl, which are particularly useful in physics: together, they form what is known as vector analysis.