In
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 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. 2D spatial directions are numerically equivalent to points on the
unit circle
and spatial directions in 3D are equivalent to a point on the
unit sphere.
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 (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. 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 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
) 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, 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. 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 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;
*
, 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, 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.
Mathematically, ...
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 variable ...
. 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 ...
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
(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:
:
:
where
is the
Kronecker delta (which is 1 for ''i'' = ''j'', and 0 otherwise) and
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
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 ...
,
is a
versor in the
3-sphere. 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
*
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
*
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,
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 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)