Editing Unit vector (section)

{{short description|Vector of length one}}
{{distinguish|Vector of ones}}
In [[mathematics]], a '''unit vector''' in a [[normed vector space]] is a [[Vector (mathematics and physics)|vector]] (often a [[vector (geometry)|spatial vector]]) of [[Norm (mathematics)|length]] 1. A unit vector is often denoted by a lowercase letter with a [[circumflex]], or "hat", as in <math>\hat{\mathbf{v}}</math> (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.,
:<math alt="u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}=(\frac{u_1}{\|\mathbf{u}\|}, \frac{u_2}{\|\mathbf{u}\|}, ... ,  \frac{u_n}{\|\mathbf{u}\|})</math>
where ‖'''u'''‖ is the [[Norm (mathematics)|norm]] (or length) of '''u''' and <math display="inline">\|\mathbf{u}\| = (u_1, u_2, ..., u_n)</math>.<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Unit Vector|url=https://mathworld.wolfram.com/UnitVector.html#:~:text=A%20unit%20vector%20is%20a,as%20the%20(finite)%20vector%20.|access-date=2020-08-19|website=Wolfram MathWorld |language=en}}</ref><ref>{{Cite web|title=Unit Vectors |url=https://brilliant.org/wiki/unit-vectors/|access-date=2020-08-19|website=Brilliant Math & Science Wiki |language=en-us}}</ref>  

The proof is the following: <math alt="u-hat equals the vector u divided by its length" display="inline">\|\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</math>

A unit vector is often used to represent [[direction (geometry)|directions]], such as [[normal direction]]s.
Unit vectors are often chosen to form the [[basis (linear algebra)|basis]] of a vector space, and every vector in the space may be written as a [[linear combination]] form of unit vectors.