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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a unit vector in a normed vector space is a vector (often a
spatial vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
) of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
1. A unit vector is often denoted by a lowercase letter with a
circumflex The circumflex () is a diacritic in the Latin and Greek scripts that is also used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from la, circumflexus "bent around" ...
, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vector'', commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and
relative direction In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
. 2D spatial directions are numerically equivalent to points on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
and spatial directions in 3D are equivalent to a point on the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac where , u, is the norm (or length) of u. The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors.


Orthogonal coordinates


Cartesian coordinates

Unit vectors may be used to represent the axes of a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
. For instance, the standard unit vectors in the direction of the ''x'', ''y'', and ''z'' axes of a three dimensional Cartesian coordinate system are : \mathbf = \begin1\\0\\0\end, \,\, \mathbf = \begin0\\1\\0\end, \,\, \mathbf = \begin0\\0\\1\end They form a set of mutually
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
unit vectors, typically referred to as a
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
in
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
. They are often denoted using common vector notation (e.g., ''i'' or \vec) rather than standard unit vector notation (e.g., \mathbf). In most contexts it can be assumed that i, j, and k, (or \vec, \vec, and \vec) are versors of a 3-D Cartesian coordinate system. The notations (\mathbf, \mathbf, \mathbf), (\mathbf_1, \mathbf_2, \mathbf_3), (\mathbf_x, \mathbf_y, \mathbf_z), or (\mathbf_1, \mathbf_2, \mathbf_3), with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
symbols such as ''i'', ''j'', ''k'', which are used to identify an element of a set or array or sequence of variables). When a unit vector in space is expressed in Cartesian notation as a linear combination of i, j, k, its three scalar components can be referred to as direction cosines. The value of each component is equal to the cosine of the angle formed by the unit vector with the respective basis vector. This is one of the methods used to describe the orientation (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis ( vector).


Cylindrical coordinates

The three
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
unit vectors appropriate to cylindrical symmetry are: * \boldsymbol (also designated \mathbf or \boldsymbol), representing the direction along which the distance of the point from the axis of symmetry is measured; * \boldsymbol, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the symmetry axis; * \mathbf, representing the direction of the symmetry axis; They are related to the Cartesian basis \hat, \hat, \hat by: : \boldsymbol = \cos(\varphi)\mathbf + \sin(\varphi)\mathbf :\boldsymbol = -\sin(\varphi) \mathbf + \cos(\varphi) \mathbf : \mathbf = \mathbf. The vectors \boldsymbol and \boldsymbol are functions of \varphi, and are ''not'' constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. The derivatives with respect to \varphi are: :\frac = -\sin \varphi\mathbf + \cos \varphi\mathbf = \boldsymbol :\frac = -\cos \varphi\mathbf - \sin \varphi\mathbf = -\boldsymbol : \frac = \mathbf.


Spherical coordinates

The unit vectors appropriate to spherical symmetry are: \mathbf, the direction in which the radial distance from the origin increases; \boldsymbol, the direction in which the angle in the ''x''-''y'' plane counterclockwise from the positive ''x''-axis is increasing; and \boldsymbol, the direction in which the angle from the positive ''z'' axis is increasing. To minimize redundancy of representations, the polar angle \theta is usually taken to lie between zero and 180 degrees. It is especially important to note the context of any ordered triplet written in
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, as the roles of \boldsymbol and \boldsymbol are often reversed. Here, the American "physics" convention is used. This leaves the
azimuthal angle An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematically, ...
\varphi defined the same as in cylindrical coordinates. The Cartesian relations are: :\mathbf = \sin \theta \cos \varphi\mathbf + \sin \theta \sin \varphi\mathbf + \cos \theta\mathbf :\boldsymbol = \cos \theta \cos \varphi\mathbf + \cos \theta \sin \varphi\mathbf - \sin \theta\mathbf :\boldsymbol = - \sin \varphi\mathbf + \cos \varphi\mathbf The spherical unit vectors depend on both \varphi and \theta, and hence there are 5 possible non-zero derivatives. For a more complete description, see
Jacobian matrix and determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
. The non-zero derivatives are: :\frac = -\sin \theta \sin \varphi\mathbf + \sin \theta \cos \varphi\mathbf = \sin \theta\boldsymbol :\frac =\cos \theta \cos \varphi\mathbf + \cos \theta \sin \varphi\mathbf - \sin \theta\mathbf= \boldsymbol :\frac =-\cos \theta \sin \varphi\mathbf + \cos \theta \cos \varphi\mathbf = \cos \theta\boldsymbol :\frac = -\sin \theta \cos \varphi\mathbf - \sin \theta \sin \varphi\mathbf - \cos \theta\mathbf = -\mathbf :\frac = -\cos \varphi\mathbf - \sin \varphi\mathbf = -\sin \theta\mathbf -\cos \theta\boldsymbol


General unit vectors

Common themes of unit vectors occur throughout
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
:


Curvilinear coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors \mathbf_n (the actual number being equal to the degrees of freedom of the space). For ordinary 3-space, these vectors may be denoted \mathbf_1, \mathbf_2, \mathbf_3. It is nearly always convenient to define the system to be orthonormal and right-handed: :\mathbf_i \cdot \mathbf_j = \delta_ :\mathbf_i \cdot (\mathbf_j \times \mathbf_k) = \varepsilon_ where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
(which is 1 for ''i'' = ''j'', and 0 otherwise) and \varepsilon_ is the Levi-Civita symbol (which is 1 for permutations ordered as ''ijk'', and −1 for permutations ordered as ''kji'').


Right versor

A unit vector in \mathbb^3 was called a right versor by W. R. Hamilton, as he developed his
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s \mathbb \subset \mathbb^4. In fact, he was the originator of the term ''vector'', as every quaternion q = s + v has a scalar part ''s'' and a vector part ''v''. If ''v'' is a unit vector in \mathbb^3, then the square of ''v'' in quaternions is –1. Thus by
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
, \exp (\theta v) = \cos \theta + v \sin \theta is a versor in the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
. When ''θ'' is a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
, the versor is a right versor: its scalar part is zero and its vector part ''v'' is a unit vector in \mathbb^3.


See also

*
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
*
Coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
* Curvilinear coordinates * Four-velocity *
Jacobian matrix and determinant In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
*
Normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
*
Polar coordinate system In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
*
Standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
*
Unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
* Unit
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
,
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
,
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
,
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, and
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
* Vector notation *
Vector of ones In mathematics, a matrix of ones or all-ones matrix is a matrix where every entry is equal to one. Examples of standard notation are given below: :J_2 = \begin 1 & 1 \\ 1 & 1 \end;\quad J_3 = \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end;\quad ...
*
Unit matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...


Notes


References

* * * {{DEFAULTSORT:Unit Vector Linear algebra Elementary mathematics 1 (number) Vectors (mathematics and physics)