A unit vector in analytic geometry is a vector of length one. In linear algebra and functional analysis , the concept of length is generalized to general vector spaces to the concept of norm . A vector in a normalized vector space , that is, a vector space on which a norm is defined, is called a unit vector or normalized vector if its norm is one.
An element of a normalized vector space is called a unit vector if applies. Unit vectors are usually indicated in applications with a hat over the variable ( ).
The elements of a basis (= basis vectors) are often chosen as unit vectors, because the use of unit vectors simplifies many calculations. For example, in Euclidean space, the standard scalar product of two unit vectors is equal to the cosine of the angle between the two.
The set of canonical unit vectors des forms an orthonormal basis with respect to the canonical scalar product , i. H. two canonical unit vectors are perpendicular to each other (= "ortho"), all are normalized (= "normal") and they form a basis.
The three canonical unit vectors of the three-dimensional vector space are also referred to in the natural sciences as :
Infinite dimensional case
In infinite-dimensional unitary vector spaces (= VR with scalar product) the (infinite) set of canonical unit vectors still forms an orthonormal system , but not necessarily a (vector space) basis . In Hilbert spaces , however, it is possible to represent every vector of the space by admitting infinite sums, which is why one speaks of an orthonormal basis .
- Principles Of Physics: A Calculus-based Text, Volume 1, Raymond A. Serway, John W. Jewett, Publisher: Cengage Learning, 2006, ISBN 9780534491437 , p. 19, limited preview in Google Book Search