A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in
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.
The notations (î, ĵ, k̂), (x̂1, x̂2, x̂3), (êx, êy, êz), or (ê1, ê2, ê3), with or without hat, are also used,[1] 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 x, y, z, 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).
The three orthogonal unit vectors appropriate to cylindrical symmetry are: They are related to the Cartesian basis
When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on.
, 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
It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of
The Cartesian relations are: The spherical unit vectors depend on both
For a more complete description, see Jacobian matrix and determinant.
The non-zero derivatives are: Common themes of unit vectors occur throughout physics and geometry:[4] A normal vector
to the plane containing and defined by the radial position vector
is necessary so that the vector equations of angular motion hold.
aligned parallel to a principal direction (red line), and a perpendicular unit vector
is in any radial direction relative to the principal line.
Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction.
In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors
[1] (the actual number being equal to the degrees of freedom of the space).
It is nearly always convenient to define the system to be orthonormal and right-handed: where
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
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
Thus the right versors extend the notion of imaginary units found in the complex plane, where the right versors now range over the 2-sphere
By extension, a right quaternion is a real multiple of a right versor.