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 "hat": {\hat{\imath}} (pronounced "i-hat"). the normalized vector or versor \mathbf{\hat{u}} of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|} 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. every vector in the space may be written as a linear combination of unit vectors. by definition, in a euclidean space the dot product of two unit vectors is the cosine of the angle between them. in three-dimensional euclidean space, the cross product of two orthogonal unit vectors is another unit vector, orthogonal to both of them. ==orthogonal coordinates== ===cartesian coordinates===  unit vectors may be used to represent the axes of a cartesian coordinate system. for instance, the unit vectors in the direction of the x, y, and z axes of a three dimensional cartesian coordinate system are :\mathbf{\hat{i}} = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{j}} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\, \mathbf{\hat{k}} = \begin{bmatrix}0\\0\\1\end{bmatrix} they are sometimes referred to as the versors of the coordinate system, and they form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra. they are often denoted using normal vector notation (e.g., i or \vec{\imath}) rather than standard unit vector notation (e.g., \mathbf{\hat{\imath}}). in most contexts it can be assumed that i, j, and k, (or \vec{\imath}, \vec{\jmath}, and \vec{k}) are versors of a 3-d cartesian coordinate system. the notations (\mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\hat{z}}), (\mathbf{\hat{x}}_1, \mathbf{\hat{x}}_2, \mathbf{\hat{x}}_3), (\mathbf{\hat{e}}_x, \mathbf{\hat{e}}_y, \mathbf{\hat{e}}_z), or (\mathbf{\hat{e}}_1, \mathbf{\hat{e}}_2, \mathbf{\hat{e}}_3), 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, used to identify an element of a set or array or sequence of variables). when a unit vector in space is expressed, with cartesian notation, as a linear combination of i, j, k, its 
1_(number) Elementary_mathematics Linear_algebra Vectors