Euclidean Vector Space

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




  1. Latin: vectus, perfect participle of vehere, "to carry"/ veho = "I carry". For historical development of the word vector, see and .


  • (Wikipedia, 2009) ⇒
    • … 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.