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 circumflex, or "hat", as in ${\displaystyle {\hat {\mathbf {v} }}}$ (pronounced "v-hat"). The term normalized vector is sometimes used as a synonym for unit vector.

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

${\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\|\mathbf {u} \|}}=({\frac {u_{1}}{\|\mathbf {u} \|}},{\frac {u_{2}}{\|\mathbf {u} \|}},...,{\frac {u_{n}}{\|\mathbf {u} \|}})}$

where ‖u‖ is the norm (or length) of u and ${\textstyle \|\mathbf {u} \|=(u_{1},u_{2},...,u_{n})}$.^[1][2]

The proof is the following: ${\textstyle \|\mathbf {\hat {u}} \|={\sqrt {{\frac {u_{1}}{\sqrt {u_{1}^{2}+...+u_{n}^{2}}}}^{2}+...+{\frac {u_{n}}{\sqrt {u_{1}^{2}+...+u_{n}^{2}}}}^{2}}}={\sqrt {\frac {u_{1}^{2}+...+u_{n}^{2}}{u_{1}^{2}+...+u_{n}^{2}}}}={\sqrt {1}}=1}$

A unit vector is often used to represent directions, such as normal directions. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors.