Unit vector

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.

definition

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 ( ). ${\displaystyle v}$ ${\displaystyle V}$ ${\displaystyle \|{\vec {v}}\|=1}$ ${\displaystyle {\hat {v}}}$

classification

A given vector that differs from the zero vector can be normalized by dividing it by its norm (= its absolute value): ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {n}}={\frac {\vec {v}}{\|{\vec {v}}\|}}}$

This vector is the unit vector pointing in the same direction as . He plays z. B. a role in the Gram-Schmidt orthogonalization method or the calculation of the Hessian normal form . ${\displaystyle {\vec {v}}}$

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.

Finite-dimensional case

Canonical unit vectors in the Euclidean plane

In the finite-dimensional real vector spaces , the most frequently preferred standard basis consists of the canonical unit vectors ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\vec {e}}_{1}={\begin{pmatrix}1\\0\\0\\\vdots \\0\end{pmatrix}},\;{\vec {e}}_{2}={\begin{pmatrix}0\\1\\0\\\vdots \\0\end{pmatrix}},\;{\vec {e}}_{3}={\begin{pmatrix}0\\0\\1\\\vdots \\0\end{pmatrix}},\;\dots ,\;{\vec {e}}_{n}={\begin{pmatrix}0\\0\\0\\\vdots \\1\end{pmatrix}}}$.

If you combine the canonical unit vectors into a matrix , you get a unit matrix .

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. ${\displaystyle \mathbb {R} ^{n}}$

example

The three canonical unit vectors of the three-dimensional vector space are also referred to in the natural sciences as : ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbf {i} ,\,\mathbf {j} ,\,\mathbf {k} }$

${\displaystyle \mathbf {i} ={\vec {e}}_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad \mathbf {j} ={\vec {e}}_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad \mathbf {k} ={\vec {e}}_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}}$

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 .

See also

Wiktionary: unit vector  - explanations of meanings, word origins, synonyms, translations

• Cartesian coordinate system

Individual evidence

1. ↑ 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

literature

• Unit vector . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).