linguistic:paraphrasing
History

In 1773, Joseph-Louis Lagrange used the component form of both the dot and cross products in order to study the tetrahedron in three dimensions.

In 1843, William Rowan Hamilton introduced the quaternion product, and with it the terms vector and scalar. Given two quaternions and, where u and v are vectors in R, their quaternion product can be summarized as. James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education.

In 1844, Hermann Grassmann published a geometric algebra not tied to dimension two or three. Grassmann developed several products, including a cross product represented then by. (See also: exterior algebra.)

In 1853, Augustin-Louis Cauchy, a contemporary of Grassmann, published a paper on algebraic keys which were used to solve equations and had the same multiplication properties as the cross product.

In 1878, William Kingdon Clifford, known for a precursor to the Clifford algebra named in his honor, published Elements of Dynamic, in which the term vector product is attested. In the book, this product of two vectors is defined to have magnitude equal to the area of the parallelogram of which they are two sides, and direction perpendicular to their plane.

In lecture notes from 1881, Gibbs represented the cross product by u × v {\displaystyle u\times v} and called it the skew product. In 1901, Gibb's student Edwin Bidwell Wilson edited and extended these lecture notes into the textbook Vector Analysis. Wilson kept the term skew product, but observed that the alternative terms cross product and vector product were more frequent.

In 1908, Cesare Burali-Forti and Roberto Marcolongo introduced the vector product notation u ∧ v. This is used in France and other areas until this day, as the symbol × {\displaystyle \times } is already used to denote multiplication and the Cartesian product.