Editing Cross product (section)

== Names and origin ==
[[File:Sarrus rule.svg|upright=1.25|thumb|right|According to [[Sarrus's rule]], the [[determinant]] of a 3×3 matrix involves multiplications between matrix elements identified by crossed diagonals]]
In 1842, [[William Rowan Hamilton]] first described the algebra of [[quaternion|quaternions]] and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with zero scalar part) is performed, it results in a quaternion with a scalar and vector part. The scalar and vector part of this Hamilton product corresponds to the negative of dot product and cross product of the two vectors.  

In 1881, [[Josiah Willard Gibbs]],<ref>{{cite book |others=Founded upon the lectures of J. William Gibbs |author=Edwin Bidwell Wilson |title=[[Vector Analysis]] |chapter=Chapter II. Direct and Skew Products of Vectors |publisher=Yale University Press |place=New Haven |year=1913}} The dot product is called "direct product", and cross product is called "skew product".</ref> and independently [[Oliver Heaviside]], introduced the notation for both the dot product and the cross product using a period ({{nowrap|1='''a''' ⋅ '''b'''}}) and an "×" ({{nowrap|1='''a''' × '''b'''}}), respectively, to denote them.<ref name=ucd>[https://www.math.ucdavis.edu/~temple/MAT21D/SUPPLEMENTARY-ARTICLES/Crowe_History-of-Vectors.pdf ''A History of Vector Analysis''] by Michael J. Crowe, Math. UC Davis.<!-- this source is typeset poorly, using "." for "⋅" and "x" for "×" --></ref>

In 1877, to emphasize the fact that the result of a dot product is a [[scalar (mathematics)|scalar]] while the result of a cross product is a [[Euclidean vector|vector]], [[William Kingdon Clifford]] coined the alternative names '''scalar product''' and '''vector product''' for the two operations.<ref name=ucd/> These alternative names are still widely used in the literature.

Both the cross notation ({{nowrap|1='''a''' × '''b'''}}) and the name '''cross product''' were possibly inspired by the fact that each [[scalar component]] of {{nowrap|1='''a''' × '''b'''}} is computed by multiplying non-corresponding components of '''a''' and '''b'''. Conversely, a dot product {{nowrap|1='''a''' ⋅ '''b'''}} involves multiplications between corresponding components of '''a''' and '''b'''. As explained [[#Matrix notation|below]], the cross product can be expressed in the form of a determinant of a special {{nowrap|3 × 3}} matrix. According to [[Sarrus's rule]], this involves multiplications between matrix elements identified by crossed diagonals.