Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in formula_1 (pronounced "v-hat"). 
The term "direction vector" is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d; 2D spatial directions represented this way are numerically equivalent to points on the unit circle.
The same construct is used to specify spatial directions in 3D, which 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.
By definition, the dot product of two unit vectors in a Euclidean space is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the cross product of two arbitrary unit vectors is a third vector orthogonal to both of them, whose length is equal to the sine of the smaller subtended angle. The normalized cross product corrects for this varying length, and yields the mutually orthogonal unit vector to the two inputs, applying the right-hand rule to resolve one of two possible directions.
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.
They are often denoted using common vector notation (e.g., i or formula_4) rather than standard unit vector notation (e.g., formula_5). In most contexts it can be assumed that i, j, and k, (or formula_6 formula_7 and formula_8) are versors of a 3-D Cartesian coordinate system. The notations formula_9, formula_10, formula_11, or formula_12, 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: 
They are related to the Cartesian basis formula_18, formula_19, formula_20 by:
It is important to note that formula_13 and formula_16 are functions of formula_28, 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 formula_29 are:
Spherical coordinates.
The unit vectors appropriate to spherical symmetry are: formula_33, the direction in which the radial distance from the origin increases; formula_34, the direction in which the angle in the "x"-"y" plane counterclockwise from the positive "x"-axis is increasing; and formula_35, the direction in which the angle from the positive "z" axis is increasing. To minimize redundancy of representations, the polar angle formula_36 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 formula_16 and formula_35 are often reversed. Here, the American "physics" convention is used. This leaves the azimuthal angle formula_39 defined the same as in cylindrical coordinates. The Cartesian relations are:
The spherical unit vectors depend on both formula_39 and formula_36, and hence there are 5 possible non-zero derivatives. For a more complete description, see Jacobian matrix and determinant. The non-zero derivatives are:
General unit vectors.
Common themes of unit vectors occur throughout physics and geometry:
Curvilinear coordinates.
In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors formula_50 (the actual number being equal to the degrees of freedom of the space). For ordinary 3-space, these vectors may be denoted formula_51. It is nearly always convenient to define the system to be orthonormal and right-handed:
where formula_54 is the Kronecker delta (which is 1 for "i" = "j", and 0 otherwise) and formula_55 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 ℝ3 was called a right versor by W. R. Hamilton, as he developed his quaternions ℍ ⊂ ℝ4. In fact, he was the originator of the term "vector", as every quaternion formula_56 has a scalar part "s" and a vector part "v". If "v" is a unit vector in ℝ3, then the square of "v" in quaternions is –1. Thus by Euler's formula, formula_57 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 ℝ3.