Unit vector

Template:Short description Template:Distinguish 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 <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 (or length) of u and <math display="inline">\|\mathbf{u}\| = (u_1, u_2, ..., u_n)</math>.<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 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.

Orthogonal coordinatesEdit

Cartesian coordinatesEdit

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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 three dimensional Cartesian coordinate system are

\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), (x̂₁, x̂₂, x̂₃), (ê_x, ê_y, ê_z), or (ê₁, ê₂, ê₃), with or without hat, are also used,<ref name=":0" /> particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with 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 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 (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis (vector).

Cylindrical coordinatesEdit

Template:See also 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 coordinatesEdit

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 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 vectorsEdit

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Common themes of unit vectors occur throughout physics and geometry:<ref>Template:Cite book</ref>

Unit vector	Nomenclature	Diagram
Tangent vector to a curve/flux line	<math> \mathbf{\hat{t}}</math>	"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 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>Template:Cite book</ref>
Parallel to some axis/line	<math> \mathbf{\hat{e}}_{\parallel} </math>	"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>	"200px" Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction.

Curvilinear coordinatesEdit

In general, a coordinate system may be uniquely specified using a number of 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-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 versorEdit

A unit vector in <math>\mathbb{R}^3</math> was called a right versor by W. R. Hamilton, as he developed his quaternions <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 units 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 Template:Math} in the complex plane.

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

NotesEdit

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