Euclidean vector

For mathematical vectors in general, see Vector (mathematics and physics). For other uses, see Vector (disambiguation).
A vector pointing from A to B

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector[1] or spatial vector[2]) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement and possibly a support, formulated as a directed line segment. A vector is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B,[3] and denoted by

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".[4] It was first used by 18th century astronomers investigating planetary revolution around the Sun.[5] 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,[6] 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: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors.[7] 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 still be 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.[8]

  1. ^ Ivanov 2001
  2. ^ Heinbockel 2001
  3. ^ Itô 1993, p. 1678; Pedoe 1988
  4. ^ Latin: vectus, perfect participle of vehere, "to carry"/ veho = "I carry". For historical development of the word vector, see "vector n.". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.) and Jeff Miller. "Earliest Known Uses of Some of the Words of Mathematics". Retrieved 2007-05-25.
  5. ^ The Oxford English Dictionary (2nd. ed.). London: Clarendon Press. 2001. ISBN 9780195219425.
  6. ^ "vector | Definition & Facts". Encyclopedia Britannica. Retrieved 2020-08-19.
  7. ^ "Vectors". www.mathsisfun.com. Retrieved 2020-08-19.
  8. ^ Weisstein, Eric W. "Vector". mathworld.wolfram.com. Retrieved 2020-08-19.