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": {\displaystyle {\hat {\imath }}} (pronounced "i-hat"). The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted asd. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are equivalent numerically to points on the unit circle.

The same construct is used to specify spatial directions in 3D. As illustrated, each unique direction is equivalent numerically to a point on the unit sphere.

Examples of two 2D direction vectorsExamples of two 3D direction vectors

The normalized vector or versor û of a non-zero vector u is the unit vector in the direction of u, i.e.,

{\displaystyle \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 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 3rd vector orthogonal to both of them having length 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.