Editing Cross product (section)

== 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.<ref>
{{cite book
 |author=Lagrange, Joseph-Louis
 |title=Oeuvres
 |volume=3
 |chapter=Solutions analytiques de quelques problèmes sur les pyramides triangulaires
 |year=1773
 |page=661
 |url=https://gallica.bnf.fr/ark:/12148/bpt6k229222d/f662.item.zoom
}}</ref><ref group=note>In modern notation, Lagrange defines <math>\mathbf{\xi} = \mathbf{y} \times \mathbf{z}</math>, <math>\boldsymbol{\eta} = \mathbf{z} \times \mathbf{x}</math>, and <math>\boldsymbol{\zeta} = \mathbf{x} \times \boldsymbol{y}</math>. Thereby, the modern <math>\mathbf{x}</math> corresponds to the three variables <math>(x, x', x'')</math> in Lagrange's notation.</ref>

In 1843, [[William Rowan Hamilton]] introduced the [[quaternion]] product, and with it the terms ''vector'' and ''scalar''. Given two quaternions {{nowrap|[0, '''u''']}} and {{nowrap|[0, '''v''']}}, where '''u''' and '''v''' are vectors in '''R'''<sup>3</sup>, their quaternion product can be summarized as {{nowrap|[−'''u''' ⋅ '''v''', '''u''' × '''v''']}}. [[James Clerk Maxwell]] used Hamilton's quaternion tools to develop his famous [[Maxwell's equations|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 {{math|[uv]}}.{{sfnp|Cajori|1929|p=[https://archive.org/details/historyofmathema00cajo_0/pages/134 134]}} (''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.{{sfnp|Crowe|1994|p=[https://archive.org/details/historyofvectora0000crow/page/83 83]}}<ref>{{cite book|last=Cauchy|first=Augustin-Louis|title=Ouvres |volume=12 |page=[https://books.google.com/books?id=0k9eAAAAcAAJ&pg=PA16 16]|year=1900}}</ref>

In 1878, [[William Kingdon Clifford]], known for a [[Geometric algebra|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.<ref>{{cite web |author-link=William Kingdon Clifford |last=Clifford |first=William Kingdon |date=1878 |url=https://archive.org/details/elementsofdynami01clifiala/page/94 |title=Elements of Dynamic, Part I |page=95 |location=London |publisher=MacMillan & Co }}</ref>

In lecture notes from 1881, [[Josiah Willard Gibbs|Gibbs]] represented the cross product by <math>u \times v</math> and called it the ''skew product''.<ref>
{{cite book
 |last1=Gibbs
 |first1=Josiah Willard
 |title=Elements of vector analysis : arranged for the use of students in physics
 |url=https://archive.org/details/elementsvectora00gibb/page/4
 |publisher=New Haven : Printed by Tuttle, Morehouse & Taylor
 |date=1884
}}
</ref>{{sfnp|Crowe|1994|p=[https://archive.org/details/historyofvectora0000crow/page/154 154]}} 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''<ref group=note>since {{math|A × B}} is read as "{{math|A}} cross {{math|B}}"</ref> and ''vector product'' were more frequent.{{sfnp|Wilson|1901|p=[https://archive.org/details/117714283/page/61 61]}}

In 1908, [[Cesare Burali-Forti]] and [[Roberto Marcolongo]] introduced the vector product notation {{math|u ∧ v}}.{{sfnp|Cajori|1929|p=[https://archive.org/details/historyofmathema00cajo_0/pages/134 134]}} This is used in [[France]] and other areas until this day, as the symbol <math>\times</math> is already used to denote [[multiplication]] and the [[Cartesian product]].{{fact|date=July 2024}}