Vector (spatial)

In mathematics, and in particular in vectorial analysis a vector is an arrow pointing from one point to another. It is characterized by a magnitude which is a scalar, a direction (which can be defined in a 3-dimensional space by the Euler angles)) and a point of application. Although it is often described by a number of "components", each of which is dependent upon the particular coordinate system being used, a vector is an object with properties which do not dependent of the coordinate system used to describe it.

Related Topics:
Mathematics - Vectorial analysis - Magnitude - Scalar - Dimension - Euler angles

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A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law.

Related Topics:
Force - Parallelogram law

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A vector can be formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below.

Related Topics:
Coordinate system - Coordinate - Tangent space - Manifold - Differential geometry

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A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus.

Related Topics:
Tensor - Four-vector - Relativity - P-vectors - Vector fields - Vector calculus

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The word vector is also now used for more general concepts (see also vector and generalizations below), but this article describes the original spatial meaning except where otherwise noted.

Related Topics:
Vector - Generalizations

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