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 "hat": (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 as d. 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.

The normalized vector or versor 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. Every vector in the space may be written as a linear combination of unit vectors.