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Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'',〔; 〕 and denoted by $\overrightarrow.$
A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier".〔Latin: vectus, perfect participle of vehere, "to carry"/ ''veho'' = "I carry". For historical development of the word ''vector'', see and (【引用サイトリンク】 url = http://jeff560.tripod.com/v.html )〕 It was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.
Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.
==History==
The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions.〔Michael J. Crowe, A History of Vector Analysis; see also his on the subject.〕
Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence. Working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane and thus erected the first space of vectors in the plane.〔
The term vector was introduced by William Rowan Hamilton as part of his system of quaternions ''q'' = ''s'' + ''v'' where "scalar" s ∈ ℝ and "vector" ''v'' ∈ ℝ3. Thus Hamilton's vectors are 3-dimensional. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments. As complex numbers use an imaginary unit to complement the real line, Hamilton considered vectors ''v'' to be the "imaginary part" of quaternions:
: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.〔W. R. Hamilton (1846) ''London, Edinburgh & Dublin Philosophical Magazine'' 3rd series 29 27〕
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. Grassmann's 1840 work ''Theorie der Ebbe und Flut'' (Theory of the Ebb and Flow) was the first system of spatial analysis 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.〔
Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 ''Elementary Treatise of Quaternions'' included extensive treatment of the nabla or 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.〔 In 1901 Edwin Bidwell Wilson published ''Vector Analysis'', adapted from Gibb's lectures, which banished any mention of quaternions in the development of vector calculus.

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