Editing Unit vector

{{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.

==Orthogonal coordinates==

===Cartesian coordinates===
{{Main|Standard basis}}

Unit vectors may be used to represent the axes of a [[Cartesian coordinate system]]. For instance, the standard ''unit vectors'' in the direction of the ''x'', ''y'', and ''z'' axes of a [[Cartesian coordinate system|three dimensional Cartesian coordinate system]] are

:<math alt="i-hat equals the 3 by 1 matrix 1,0,0; j-hat equals the 3 by 1 matrix 0,1,0; k-hat equals the 3 by 1 matrix 0,0,1">
\mathbf{\hat{x}} = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{y}} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\,  \mathbf{\hat{z}} = \begin{bmatrix}0\\0\\1\end{bmatrix}</math>

They form a set of mutually [[orthogonal]] ''unit vectors'', typically referred to as a [[standard basis]] in [[linear algebra]].

They are often denoted using common [[vector notation]] (e.g., '''x''' or <math alt="vector i">\vec{x}</math>) rather than standard unit vector notation (e.g., '''x̂'''). In most contexts it can be assumed that '''x''', '''y''', and '''z''', (or <math alt="vector i">\vec{x},</math> <math alt="vector j">\vec{y},</math> and <math alt="vector k"> \vec{z}</math>) are versors of a 3-D [[Cartesian coordinate system]]. The notations ('''î''', '''ĵ''', '''k̂'''), ('''x̂<sub>1</sub>''', '''x̂<sub>2</sub>''', '''x̂<sub>3</sub>'''), ('''ê<sub>x</sub>''', '''ê<sub>y</sub>''', '''ê<sub>z</sub>'''), or ('''ê<sub>1</sub>''', '''ê<sub>2</sub>''', '''ê<sub>3</sub>'''), with or without [[Hat notation|hat]], are also used,<ref name=":0" /> particularly in contexts where '''i''', '''j''', '''k''' might lead to confusion with another quantity (for instance with [[Indexed family|index]] symbols such as ''i'', ''j'', ''k'', which are used to identify an element of a set or array or sequence of variables).

When a unit vector in space is expressed in [[Cartesian coordinate system#Representing a vector with Cartesian notation|Cartesian notation]] as a linear combination of '''x''', '''y''', '''z''', its three scalar components can be referred to as [[direction cosines]]. The value of each component is equal to the cosine of the angle formed by the unit vector with the respective basis vector. This is one of the methods used to describe the [[Orientation (mathematics)|orientation]] (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis ([[vector (geometry)|vector]]).

===Cylindrical coordinates===
{{see also|Jacobian matrix}}
The three [[orthogonal]] unit vectors appropriate to cylindrical symmetry are: 
* <math alt="rho-hat">\boldsymbol{\hat{\rho}}</math> (also designated <math alt="e-hat">\mathbf{\hat{e}}</math> or <math alt="s-hat">\boldsymbol{\hat s}</math>), representing the direction along which the distance of the point from the axis of symmetry is measured; 
* <math alt="phi-hat">\boldsymbol{\hat \varphi}</math>, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the [[symmetry axis]];
* <math alt="z-hat">\mathbf{\hat{z}}</math>, representing the direction of the symmetry axis; 
They are related to the Cartesian basis <math alt="x-hat">\hat{x}</math>, <math alt="y-hat">\hat{y}</math>, <math alt="z-hat">\hat{z}</math> by:

:<math alt="rho-hat equals cosine of phi in the x-hat direction plus sine of phi in the y-hat direction"> \boldsymbol{\hat{\rho}} = \cos(\varphi)\mathbf{\hat{x}} + \sin(\varphi)\mathbf{\hat{y}}</math>
:<math alt="phi-hat equals negative sine of phi in the x-hat direction plus the cosine of phi in the y-hat direction">\boldsymbol{\hat \varphi} = -\sin(\varphi) \mathbf{\hat{x}} + \cos(\varphi) \mathbf{\hat{y}}</math>
:<math alt="z-hat equals z-hat"> \mathbf{\hat{z}} = \mathbf{\hat{z}}.</math>

The vectors <math alt="rho-hat">\boldsymbol{\hat{\rho}}</math> and <math alt="phi-hat">\boldsymbol{\hat \varphi}</math> are functions of <math alt="coordinate phi">\varphi,</math> and are ''not'' constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. The derivatives with respect to <math>\varphi</math> are:

:<math alt="partial derivative of rho-hat with respect to phi equals minus sine of phi in the x-hat direction plus cosine of phi in the y-hat direction equals phi-hat">\frac{\partial \boldsymbol{\hat{\rho}}} {\partial \varphi} = -\sin \varphi\mathbf{\hat{x}} + \cos \varphi\mathbf{\hat{y}} = \boldsymbol{\hat \varphi}</math>
:<math alt="partial derivative of phi-hat with respect to phi equals minus cosine of phi in the x-hat direction minus sine of phi in the y-hat direction equals minus rho-hat">\frac{\partial \boldsymbol{\hat \varphi}} {\partial \varphi} = -\cos \varphi\mathbf{\hat{x}} - \sin \varphi\mathbf{\hat{y}} = -\boldsymbol{\hat{\rho}}</math>
:<math alt="partial derivative of z-hat with respect to phi equals zero"> \frac{\partial \mathbf{\hat{z}}} {\partial \varphi} = \mathbf{0}.</math>

===Spherical coordinates===

The unit vectors appropriate to spherical symmetry are: <math alt="r-hat">\mathbf{\hat{r}}</math>, the direction in which the radial distance from the origin increases; <math alt="phi-hat">\boldsymbol{\hat{\varphi}}</math>, the direction in which the angle in the ''x''-''y'' plane counterclockwise from the positive ''x''-axis is increasing; and <math alt="theta-hat">\boldsymbol{\hat \theta}</math>, the direction in which the angle from the positive ''z'' axis is increasing. To minimize redundancy of representations, the polar angle <math alt="theta">\theta</math> is usually taken to lie between zero and 180 degrees. It is especially important to note the context of any ordered triplet written in [[spherical coordinates]], as the roles of <math alt="phi-hat">\boldsymbol{\hat \varphi}</math> and <math alt="theta-hat">\boldsymbol{\hat \theta}</math> are often reversed. Here, the American "physics" convention<ref>Tevian Dray and Corinne A. Manogue, Spherical Coordinates, College Math Journal 34, 168-169 (2003).</ref> is used. This leaves the [[azimuthal angle]] <math alt="phi">\varphi</math> defined the same as in cylindrical coordinates. The [[Cartesian coordinate system|Cartesian]] relations are:

:<math alt="r-hat equals sin of theta times cosine of phi in the x-hat direction plus sine of theta times sine of phi in the y-hat direction plus cosine of theta in the z-hat direction">\mathbf{\hat{r}} = \sin \theta \cos \varphi\mathbf{\hat{x}}  + \sin \theta \sin \varphi\mathbf{\hat{y}} + \cos \theta\mathbf{\hat{z}}</math>

:<math alt="theta-hat equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction">\boldsymbol{\hat \theta} = \cos \theta \cos \varphi\mathbf{\hat{x}} + \cos \theta \sin \varphi\mathbf{\hat{y}} - \sin \theta\mathbf{\hat{z}}</math>

:<math alt="phi-hat equals minus sine of phi in the x-hat direction plus cosine of phi in the y-hat direction">\boldsymbol{\hat \varphi} = - \sin \varphi\mathbf{\hat{x}} + \cos \varphi\mathbf{\hat{y}}</math>

The spherical unit vectors depend on both <math alt="phi">\varphi</math> and <math alt="theta">\theta</math>, and hence there are 5 possible non-zero derivatives. For a more complete description, see [[Jacobian matrix and determinant]]. The non-zero derivatives are:

:<math alt="partial derivative of r-hat with respect to phi equals minus sine of theta times sine of phi in the x-hat direction plus sine of theta times cosine of phi in the y-hat direction equals sine of theta in the phi-hat direction">\frac{\partial \mathbf{\hat{r}}} {\partial \varphi} = -\sin \theta \sin \varphi\mathbf{\hat{x}} + \sin \theta \cos \varphi\mathbf{\hat{y}} = \sin \theta\boldsymbol{\hat \varphi}</math>

:<math alt="partial derivative of r-hat with respect to theta equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction equals theta-hat">\frac{\partial \mathbf{\hat{r}}} {\partial \theta} =\cos \theta \cos \varphi\mathbf{\hat{x}} + \cos \theta \sin \varphi\mathbf{\hat{y}} - \sin \theta\mathbf{\hat{z}}= \boldsymbol{\hat \theta}</math>

:<math alt="partial derivative of theta-hat with respect to phi equals minus cosine of theta times sine of phi in the x-hat direction plus cosine of theta times cosine of phi in the y-hat direction equals cosine of theta in the phi-hat direction">\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \varphi} =-\cos \theta \sin \varphi\mathbf{\hat{x}} + \cos \theta \cos \varphi\mathbf{\hat{y}} = \cos \theta\boldsymbol{\hat \varphi}</math>

:<math alt="partial derivative of theta-hat with respect to theta equals minus sine of theta times cosine of phi in the x-hat direction minus sine of theta times sine of phi in the y-hat direction minus cosine of theta in the z-hat direction equals minus r-hat">\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \theta} = -\sin \theta \cos \varphi\mathbf{\hat{x}} - \sin \theta \sin \varphi\mathbf{\hat{y}} - \cos \theta\mathbf{\hat{z}} = -\mathbf{\hat{r}}</math>

:<math alt="partial derivative of phi-hat with respect to phi equals minus cosine of phi in the x-hat direction minus sine of phi in the y-hat direction equals minus sine of theta in the r-hat direction minus cosine of theta in the theta-hat direction">\frac{\partial \boldsymbol{\hat{\varphi}}} {\partial \varphi} = -\cos \varphi\mathbf{\hat{x}} - \sin \varphi\mathbf{\hat{y}} = -\sin \theta\mathbf{\hat{r}} -\cos \theta\boldsymbol{\hat{\theta}}</math>

===General unit vectors===

{{main|Orthogonal coordinates}}

Common themes of unit vectors occur throughout [[physics]] and [[geometry]]:<ref>{{cite book|title=Calculus (Schaum's Outlines Series)|edition=5th|publisher=Mc Graw Hill|author1=F. Ayres |author2=E. Mendelson |year=2009|isbn=978-0-07-150861-2}}</ref>

{| class="wikitable"
|-

! scope="col" width="200" | Unit vector
! scope="col" width="150" | Nomenclature
! scope="col" width="410" | Diagram
|-
| Tangent vector to a curve/flux line || <math> \mathbf{\hat{t}}</math> || rowspan="3" | [[File:Tangent normal binormal unit vectors.svg|200px|"200px"]] [[File:Polar coord unit vectors and normal.svg|200px|"200px"]]
A normal vector <math> \mathbf{\hat{n}} </math> to the plane containing and defined by the radial position vector <math> r \mathbf{\hat{r}} </math> and angular tangential direction of rotation <math> \theta \boldsymbol{\hat{\theta}} </math> is necessary so that the vector equations of angular motion hold.
|-
|Normal to a surface tangent plane/plane containing radial position component and angular tangential component
|| <math> \mathbf{\hat{n}}</math>

In terms of [[spherical coordinate system|polar coordinates]];
<math> \mathbf{\hat{n}} = \mathbf{\hat{r}} \times \boldsymbol{\hat{\theta}} </math>
|-
| Binormal vector to tangent and normal
|| <math> \mathbf{\hat{b}} = \mathbf{\hat{t}} \times \mathbf{\hat{n}} </math><ref>{{cite book|title=Vector Analysis (Schaum's Outlines Series)|edition=2nd|publisher=Mc Graw Hill|author1=M. R. Spiegel |author2=S. Lipschutz |author3=D. Spellman |year=2009|isbn=978-0-07-161545-7}}</ref>
|-
| Parallel to some axis/line || <math> \mathbf{\hat{e}}_{\parallel} </math> || rowspan="2" | [[File:Perpendicular and parallel unit vectors.svg|200px|"200px"]]
One unit vector <math> \mathbf{\hat{e}}_{\parallel}</math> aligned parallel to a principal direction (red line), and a perpendicular unit vector <math> \mathbf{\hat{e}}_{\bot}</math> is in any radial direction relative to the principal line.
|-
| Perpendicular to some axis/line in some radial direction
|| <math> \mathbf{\hat{e}}_{\bot} </math>
|-
| Possible angular deviation relative to some axis/line
|| <math> \mathbf{\hat{e}}_{\angle} </math>
|| [[File:Angular unit vector.svg|200px|"200px"]]
Unit vector at acute deviation angle ''φ'' (including 0 or ''π''/2 rad) relative to a principal direction.
|-
|}

==Curvilinear coordinates==
In general, a coordinate system may be uniquely specified using a number of [[Linear independence|linearly independent]] unit vectors <math alt="e-hat sub n">\mathbf{\hat{e}}_n</math><ref name=":0" /> (the actual number being equal to the degrees of freedom of the space). For ordinary 3-space, these vectors may be denoted <math alt="e-hat sub 1, e-hat sub 2, e-hat sub 3">\mathbf{\hat{e}}_1, \mathbf{\hat{e}}_2, \mathbf{\hat{e}}_3</math>. It is nearly always convenient to define the system to be orthonormal and [[Right-hand rule|right-handed]]:

:<math alt="e-hat sub i dot e-hat sub j equals Kronecker delta of i and j">\mathbf{\hat{e}}_i \cdot \mathbf{\hat{e}}_j = \delta_{ij} </math>
:<math alt="e-hat sub i dot e-hat sub j cross e-hat sub k = epsilon sub ijk">\mathbf{\hat{e}}_i \cdot (\mathbf{\hat{e}}_j \times \mathbf{\hat{e}}_k) = \varepsilon_{ijk} </math>

where <math> \delta_{ij} </math> is the [[Kronecker delta]] (which is 1 for ''i'' = ''j'', and 0 otherwise) and  <math alt="epsilon sub i,j,k"> \varepsilon_{ijk} </math> is the [[Levi-Civita symbol]] (which is 1 for permutations ordered as ''ijk'', and −1 for permutations ordered as ''kji'').

==Right versor==
A unit vector in <math>\mathbb{R}^3</math> was called a '''right versor''' by [[W. R. Hamilton]], as he developed his [[quaternion]]s <math>\mathbb{H} \subset \mathbb{R}^4</math>. In fact, he was the originator of the term ''vector'', as every quaternion <math>q = s + v</math> has a scalar part ''s'' and a vector part ''v''. If ''v'' is a unit vector in <math>\mathbb{R}^3</math>, then the square of ''v'' in quaternions is −1. Thus by [[Euler's formula]], <math>\exp (\theta v) = \cos \theta + v \sin \theta</math> is a [[versor]] in the [[3-sphere]]. When ''θ'' is a [[right angle]], the versor is a right versor: its scalar part is zero and its vector part ''v'' is a unit vector in <math>\mathbb{R}^3</math>.

Thus the right versors extend the notion of [[imaginary unit]]s found in the [[complex plane]], where the right versors now range over the [[2-sphere]] <math>\mathbb{S}^2 \subset \mathbb{R}^3 \subset \mathbb{H} </math> rather than the pair {{math|{''i'', −''i''}}} in the complex plane.

By extension, a '''right quaternion''' is a real multiple of a right versor.

==See also==
{{wiktionary|unit vector}}
*[[Cartesian coordinate system]]
*[[Coordinate system]]
*[[Curvilinear coordinates]]
*[[Four-velocity]]
*[[Jacobian matrix and determinant]]
*[[Normal vector]]
*[[Polar coordinate system]]
*[[Standard basis]]
*[[Unit interval]]
* Unit [[unit square|square]], [[unit cube|cube]], [[unit circle|circle]], [[unit sphere|sphere]], and [[unit hyperbola|hyperbola]]
* [[Vector notation]]
*[[Vector of ones]]
*[[Unit matrix]]

==Notes==
{{Reflist}}

==References==
*{{cite book|author1=G. B. Arfken  |author2=H. J. Weber |name-list-style=amp |title=Mathematical Methods for Physicists|edition=5th|year=2000|publisher=Academic Press|isbn=0-12-059825-6}}
*{{cite book|first=Murray R.|last=Spiegel|title=Schaum's Outlines: Mathematical Handbook of Formulas and Tables|edition=2nd|year=1998|publisher=McGraw-Hill|isbn=0-07-038203-4}}
*{{cite book|first=David J.|last=Griffiths|title=Introduction to Electrodynamics|edition=3rd|year=1998|publisher=Prentice Hall|isbn=0-13-805326-X|url-access=registration|url=https://archive.org/details/introductiontoel00grif_0}}

{{DEFAULTSORT:Unit Vector}}
[[Category:Linear algebra]]
[[Category:Elementary mathematics]]
[[Category:1 (number)]]
[[Category:Vectors (mathematics and physics)]]