In
mathematics, a unit vector in a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
is a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
(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 ...
) of
length 1. A unit vector is often denoted by a lowercase letter with a
circumflex, or "hat", as in
(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 ...
.
![2D Direction Vectors](https://upload.wikimedia.org/wikipedia/commons/d/d9/2D_Direction_Vectors.svg)
The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,
:
where , u, is the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
(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
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
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. For instance, the standard unit vectors in the direction of the ''x'', ''y'', and ''z'' axes of a three dimensional Cartesian coordinate system are
:
They form a set of mutually
orthogonal unit vectors, typically referred to as a
standard basis 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 matrices ...
.
They are often denoted using common vector notation (e.g., ''i'' or
) rather than standard unit vector notation (e.g.,
). In most contexts it can be assumed that i, j, and k, (or
and
) are versors of a 3-D Cartesian coordinate system. The notations
,
,
, or
, with or without
hat
A hat is a head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorporate mecha ...
, are also used,
particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with
index 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
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to a ...
. 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
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
(angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis (
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
).
Cylindrical coordinates
The three
orthogonal unit vectors appropriate to cylindrical symmetry are:
*
(also designated
or
), representing the direction along which the distance of the point from the axis of symmetry is measured;
*
, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the
symmetry axis
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
;
*
, representing the direction of the symmetry axis;
They are related to the Cartesian basis
,
,
by:
:
:
:
The vectors
and
are functions of
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
are:
:
:
:
Spherical coordinates
The unit vectors appropriate to spherical symmetry are:
, the direction in which the radial distance from the origin increases;
, the direction in which the angle in the ''x''-''y'' plane counterclockwise from the positive ''x''-axis is increasing; and
, the direction in which the angle from the positive ''z'' axis is increasing. To minimize redundancy of representations, the polar angle
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
and
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.
Mathematicall ...
defined the same as in cylindrical coordinates. The
Cartesian relations are:
:
:
:
The spherical unit vectors depend on both
and
, 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 variables ...
. The non-zero derivatives are:
:
:
:
:
:
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 r ...
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 ...
:
Curvilinear coordinates
In general, a coordinate system may be uniquely specified using a number of
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
unit vectors
(the actual number being equal to the degrees of freedom of the space). For ordinary 3-space, these vectors may be denoted
. It is nearly always convenient to define the system to be orthonormal and
right-handed
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...
:
:
:
where
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
is the
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
(which is 1 for permutations ordered as ''ijk'', and −1 for permutations ordered as ''kji'').
Right versor
A unit vector in
was called a right versor by
W. R. Hamilton, as he developed his
quaternions
. In fact, he was the originator of the term ''vector'', as every quaternion
has a scalar part ''s'' and a vector part ''v''. If ''v'' is a unit vector in
, 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 fo ...
,
is a
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
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, the versor is a right versor: its scalar part is zero and its vector part ''v'' is a unit vector in
.
See also
*
Cartesian coordinate system
*
Coordinate system
*
Curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
*
Four-velocity
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
*
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 variables ...
*
Normal vector
*
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
*
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,
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 con ...
,
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 th ...
, 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
In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
For representing a vector, the common typographic convention is l ...
*
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;\qu ...
*
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)