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 v ^ {\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., u ^ = u ‖ u ‖ = ( u 1 ‖ u ‖ , u 2 ‖ u ‖ , . . . , u n ‖ u ‖ ) {\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\|\mathbf {u} \|}}=\left({\frac {u_{1}}{\|\mathbf {u} \|}},{\frac {u_{2}}{\|\mathbf {u} \|}},...,{\frac {u_{n}}{\|\mathbf {u} \|}}\right)} where ‖u‖ is the norm (or length) of u and u = ( u 1 , u 2 , . . . , u n ) {\textstyle \mathbf {u} =(u_{1},u_{2},...,u_{n})} . The proof is the following: ‖ u ^ ‖ = u 1 u 1 2 + . . . + u n 2 2 + . . . + u n u 1 2 + . . . + u n 2 2 = u 1 2 + . . . + u n 2 u 1 2 + . . . + u n 2 = 1 = 1 {\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.