Home All Definitions Pre-Calculus Vector Definition

Vector Definition

Vectors are a quantity, drawn as an arrow, with both direction and magnitude. For example, force and velocity are vectors. If a quantity has a magnitude but no direction, it is referred to as a scalar. Temperature, length, and mass are examples of scalars. Five kilometers east is an example of a vector whereas just 5 kilometers would mean a scalar.

In mathematics and physics, a vector is an element of a vector space. For many specific vector spaces, the vectors have received specific names, which are listed below. Historically, vectors were introduced in geometry and physics (typically in mechanics) before the formalization of the concept of vector space. Therefore, one talks often of vectors without specifying the vector space to which they belong. Specifically, in a Euclidean space, one considers spatial vectors, also called Euclidean vectors which are used to represent quantities that have both magnitude and direction, and may be added and scaled (that is multiplied by a real number) for forming a vector space.

Vectors in Specific Vector Spaces

List of vectors in specific vector spaces:

  • Column vector, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.

  • Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.

  • Coordinate vector, the n-tuple of the coordinates of a vector on a basis of n elements. For a vector space over a field F, these n-tuples form the vector space Fn (where the operation are pointwise addition and scalar multiplication).

  • Displacement vector, a vector that specifies the change in position of a point relative to a previous position. Displacement vectors belong to the vector space of translations.

  • Position vector of a point, the displacement vector from a reference point (called the origin) to the point. A position vector represents the position of a point in a Euclidean space or an affine space.

  • Velocity vector, the derivative, with respect to time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations.

  • Pseudovector, also called axial vector, an element of the dual of a vector space. In a inner product space, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a pseudo vector from a vector. The distinction becomes apparent when one changes coordinates: the matrix used for a change of coordinates of pseudovectors is the transpose of that of vectors.

  • Tangent vector, an element of the tangent space of a curve, a surface or, more generally, a differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space)

  • Normal vector or simply normal, in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point. Normals are pseudovectors that belong to the dual of the tangent space.

  • Gradient, the coordinates vector of the partial derivatives of a function of several real variables. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a pseudo vector that is normal to a level curve.

  • Four-vector, in the theory of relativity, a vector in a four-dimensional real vector space called Minkowski space

Related Definitions

Sources

“Vector (Mathematics and Physics).” Wikipedia, Wikimedia Foundation, 24 Mar. 2020, en.wikipedia.org/wiki/Vector_(mathematics_and_physics).

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