Euclidean Vector Space

A Euclidean Vector Space is an Hilbert metric space of Euclidean vectors (positive, definite) with an Euclidean distance function.

• Context:
  • It can (typically) have a Translation Operation.
  • It can (typically) have a Rotation Operation.
  • It can (typically) have an Angle Function.
  • It can have an Euclidean Subspace.
  • It can contain an Euclidean Space Object.
  • It has X? relation with an Inner Product Space.
  • It can range from being a 2-Dimensional Euclidean Space, to being a 3-Dimensional Euclidean Space, to being a 4-Dimensional Euclidean Space to being ...
  • …
• Counter-Example(s):
  • a Real Vector Metric Space.
  • a Minkowski Space, with an origin point.
• See: Axis, Vector, 3-Dimensional Space, 2-Dimensional Space, Euclidean Geometry.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Euclidean_space Retrieved:2015-2-7.
  • In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are also used to define rational numbers. When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. It means that points of the space are specified with collections of real numbers, and geometric shapes are defined as equations and inequalities. This approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the real coordinate space ('Rn) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the -dimensional Euclidean space by Eⁿ if they wish to emphasize its Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Euclidean_vector Retrieved:2014-9-27.
  • 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 [math]\displaystyle{ \overrightarrow{AB}. }[/math] A vector is what is needed to "carry" the point A to the point B ; the Latin word vector means "carrier". 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.

2009

• http://en.wiktionary.org/wiki/Euclidean_space
  • 1. Ordinary two- or three-dimensional space, characterised by an infinite extent along each dimension and a constant distance between any pair of parallel lines.
  • 2. (mathematics) any vector space on which a real-valued inner product is defined.
• (WordNet, 2009) ⇒ http://wordnetweb.princeton.edu/perl/webwn?s=euclidean%20space
  • S: (n) Euclidean space (a space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional)

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Euclidean_space
  • … An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean. For example, the surface of a sphere is not; a triangle on a sphere (suitably defined) will have angles that sum to something greater than 180 degrees. In fact, there is essentially only one Euclidean space of each dimension, while there are many non-Euclidean spaces of each dimension. Often these other spaces are constructed by systematically deforming Euclidean space.
  • One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation about a fixed point in the plane, in which every point in the plane turns about that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (that is, subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections. (See Euclidean group.)
  • In order to make all of this mathematically precise, one must clearly define the notions of distance, angle, translation, and rotation. The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. For then:
    • the vectors in the vector space correspond to the points of the Euclidean plane,
    • the addition operation in the vector space corresponds to translation, and
    • the inner product implies notions of angle and distance, which can be used to define rotation.

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