Unit vector

In that is in Unit vector or standardized vector Vector with that Standard (descriptive: that two unit vectors equal the cosine of the angle between the two.

Finite-dimensional case

Into the finite-dimensional real Vector spaces <math>\R^n</math> the most frequently preferential exists Standard basis from that canonical unit vectors

<math>

e_1 = \begin{pmatrix} 1 \ \ 0 \ \ 0 \ \ \vdots \ \ 0 \end{pmatrix}, \; e_2 = \begin{pmatrix} 0 \ \ 1 \ \ 0 \ \ \vdots \ \ 0 \end{pmatrix}, \; e_3 = \begin{pmatrix} 0 \ \ 0 \ \ 1 \ \ \vdots \ \ 0 \end{pmatrix}, \; \dots, \; e_n = \begin{pmatrix} 0 \ \ 0 \ \ 0 \ \ \vdots \ \ 1 \end{pmatrix} </math>.

One seizes the canonical unit vectors to one Matrix together, one receives one Unit matrix.

The quantity of the canonical unit vectors <math>\R^n</math> forms relative for the canonical one Orthonormal basis, D.h. two canonical unit vectors each stand perpendicularly one on the other (= "ortho"), all are standardized (= "normally") and they form a basis.

Example

The three canonical unit vectors of the three-dimensional vector space <math>\R^3</math> sometimes become in the applied natural sciences with <math>\mathbf{i}, \, \mathbf{j}, \, \mathbf{k}</math> designated:

<math>

\mathbf{i} = e_1 = \begin{pmatrix}1\\0\\0\end{pmatrix}, \quad \mathbf{j} = e_2 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad \mathbf{k} = e_3 = \begin{pmatrix}0\\0\\1\end{pmatrix} </math>

Infinite-dimensional case

In infinite-dimensional unitary vector spaces (= VR with dot product) (infinite) the quantity of the canonical unit vectors still imagines Orthonormal system, but not necessarily one (vector space)Basis. In Hilbertraeumen succeeds however by permission of infinite sums representing each vector of the area one continues to speak therefore of one Orthonormal basis.

See also: Cartesian coordinate system