# Unit Vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a "hat", like this: (pronounced "i-hat").

In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1.

The normalized vector or versor of a non-zero vector u is the unit vector codirectional with 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.

The elements of a basis are usually chosen to be unit vectors. Every vector in the space may be written as a linear combination of unit vectors. The most commonly encountered bases are Cartesian, polar, and spherical coordinates. Each uses different unit vectors according to the symmetry of the coordinate system. Since these systems are encountered in so many different contexts, it is not uncommon to encounter different naming conventions than those used here.

### Other articles related to "unit vector, vector, unit vectors, vectors":

Lambdavacuum Solution - Einstein Tensor
... A frame consists of four unit vector fields Here, the first is a timelike unit vector field and the others are spacelike unit vector fields, and is everywhere orthogonal to the world lines of a family of observers (not ...
Classical Hamiltonian Quaternions - Classical Elements of A Quaternion - Vector - Unit Vector
... A unit vector is a vector of length one ... Examples of unit vectors include i, j and k ...
Spinors In Three Dimensions - Examples in Physics - Spinors of The Pauli Spin Matrices
... The Pauli matrices are a vector of three 2×2 matrices that are used as spin operators ... Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the ... that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector ...
Unit Vector - Curvilinear Coordinates
... system may be uniquely specified using a number of linearly independent unit vectors equal to the degrees of freedom of the space ... For ordinary 3-space, these vectors may be denoted ...
Born Coordinates - Langevin Observers in The Cylindrical Chart
... stationary (inertial) observers Here, is a timelike unit vector field while the others are spacelike unit vector fields at each event, all four are mutually orthogonal and determine the infinitesimal Lorentz frame ... Each integral curve of the timelike unit vector field appears in the cylindrical chart as a helix with constant radius (such as the red curve in the figure at right) ... figure at right) of the spacelike basis vector, we obtain a curve which we might hope can be interpreted as a "line of simultaneity" for the ring-riding observers ...

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