Unit vector

Is a unit vector in the analytical geometry, a vector of length one. In linear algebra and functional analysis of the concept of length is generalized to general vector spaces the notion of norm. A vector in a normalized vector space, that is a vector space, on which a standard has been defined, is a unit vector and a normalized vector, if its norm is one.

Definition

An element of a normed vector space is called a unit vector if and only if. Unit vectors are usually marked in the applications with a hat over the variable ().

Classification

A given, different from the zero vector vector can be standardized by dividing it by its norm ( = its amount):

This vector is the unit vector as shown in the same direction. He plays a role in the Gram-Schmidt orthogonalization 's or the calculation of the Hessian normal form, for example.

The elements of a base ( = basis vectors ) are often chosen as unit vectors, because by the use of unit vectors many bills to be simplified. For example, in a Euclidean space, the standard scalar product of two unit vectors is equal to the cosine of the angle between the two.

Finite-dimensional case

In the finite-dimensional real vector spaces, the most preferred standard basis consists of the canonical unit vectors

Summarizing the canonical unit vectors in a matrix together, we obtain an identity matrix.

The set of canonical unit vectors of the form with respect to the canonical scalar product of an orthonormal basis, ie two canonical unit vectors perpendicular to each other ( = " ortho " ), all are normalized ( = "normal " ), and they form a basis.

Example

The three canonical unit vectors of the three-dimensional vector space are sometimes referred to in applied science with:

Infinite-dimensional case

In the infinite-dimensional unitary vector spaces ( = VR with scalar product ), the ( infinite) set of canonical unit vectors is still an orthonormal system, but not necessary a ( vector space ) basis. In Hilbert spaces, however, it succeeds by allowing infinite sums represent each vector of the space, we therefore further speaks of an orthonormal basis.