Editing Euclidean vector (section)

==History==
The vector concept, as it is known today, is the result of a gradual development over a period of more than 200 years. About a dozen people contributed significantly to its development.<ref name="Crowe">[[Michael J. Crowe]], [[A History of Vector Analysis]]; see also his {{cite web |url=http://www.nku.edu/~curtin/crowe_oresme.pdf |title=lecture notes |access-date=2010-09-04 |url-status=dead |archive-url=https://web.archive.org/web/20040126161844/http://www.nku.edu/~curtin/crowe_oresme.pdf |archive-date=January 26, 2004 }} on the subject.</ref> In 1835, [[Giusto Bellavitis]] abstracted the basic idea when he established the concept of [[equipollence (geometry)|equipollence]]. Working in a Euclidean plane, he made equipollent any pair of [[parallel (geometry)|parallel]] line segments of the same length and orientation. Essentially, he realized an [[equivalence relation]] on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane.<ref name="Crowe"/>{{rp|52–4}} The term ''vector'' was introduced by [[William Rowan Hamilton]] as part of a [[quaternion]], which is a sum {{math|1=''q'' = ''s'' + ''v''}} of a [[real number]] {{math|''s''}} (also called ''scalar'') and a 3-dimensional ''vector''. Like Bellavitis, Hamilton viewed vectors as representative of [[equivalence class|classes]] of equipollent directed segments. As [[complex number]]s use an [[imaginary unit]] to complement the [[real line]], Hamilton considered the vector {{math|''v''}} to be the ''imaginary part'' of a quaternion:<ref>W. R. Hamilton (1846) ''London, Edinburgh & Dublin Philosophical Magazine'' 3rd series 29 27</ref>

{{blockquote|The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.}}

Several other mathematicians developed vector-like systems in the middle of the nineteenth century,  including [[Augustin Cauchy]], [[Hermann Grassmann]], [[August Möbius]], [[Comte de Saint-Venant]], and [[Matthew O'Brien (mathematician)|Matthew O'Brien]]. Grassmann's 1840 work ''Theorie der Ebbe und Flut'' (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s.<ref name="Crowe"/> [[Peter Guthrie Tait]] carried the quaternion standard after Hamilton. His 1867 ''Elementary Treatise of Quaternions'' included extensive treatment of the nabla or [[del|del operator]] ∇. In 1878, ''[[Elements of Dynamic]]'' was published by [[William Kingdon Clifford]]. Clifford simplified the quaternion study by isolating the [[dot product]] and [[cross product]] of two vectors from the complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth.

[[Josiah Willard Gibbs]], who was exposed to quaternions through [[James Clerk Maxwell]]'s ''Treatise on Electricity and Magnetism'', separated off their vector part for independent treatment. The first half of Gibbs's ''Elements of Vector Analysis'', published in 1881, presents what is essentially the modern system of vector analysis.<ref name="Crowe" /><ref name=":1" /> In 1901, [[Edwin Bidwell Wilson]] published ''[[Vector Analysis]]'', adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus.