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In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a "hat", like this: $\hat{ı}$ (pronounced "i-hat").

The normalized vector or versor $\hat{𝐮}$ of a non-zero vector u is the unit vector codirectional with u, i.e.,

\hat{𝐮} = \frac{𝐮}{‖ 𝐮 ‖}

where ||u|| is the norm (or length) of u. The term normalized vector is sometimes used as a synonym for unit vector.

The elements of a basis are usually chosen to be unit vectors. Every vector in the space may be written as a linear combination of unit vectors. The most commonly encountered bases are Cartesian, polar, and spherical coordinates. Each uses different unit vectors according to the symmetry of the coordinate system. Since these systems are encountered in so many different contexts, it is not uncommon to encounter different naming conventions than those used here.

By definition, in Euclidean space the dot product of two unit vectors is simply the cosine of the angle between them. In three-dimensional Euclidean space, the cross product of two orthogonal unit vectors is another unit vector, orthogonal to both of them.

Orthogonal coordinates

Cartesian coordinates

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Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the unit vectors codirectional with the x, y, and z axes of a three dimensional Cartesian coordinate system are

\hat{ı} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], \hat{ȷ} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], \hat{𝐤} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]

They are sometimes referred to as the versors of the coordinate system, and they form a set of mutually orthogonal unit vectors which, in the context of linear algebra, is typically referred to as a standard basis.

They are often denoted using normal vector notation (e.g., i or $\vec{ı}$ ) rather than standard unit vector notation (e.g., $\hat{ı}$ ). In most contexts it can be assumed that i, j, and k, (or $\vec{ı},$ $\vec{ȷ},$ and $\vec{k}$ ) are versors of a 3-D Cartesian coordinate system. The notations $(\hat{𝐱}, \hat{𝐲}, \hat{𝐳})$ , $({\hat{𝐱}}_{1}, {\hat{𝐱}}_{2}, {\hat{𝐱}}_{3})$ , $({\hat{𝐞}}_{x}, {\hat{𝐞}}_{y}, {\hat{𝐞}}_{z})$ , or $({\hat{𝐞}}_{1}, {\hat{𝐞}}_{2}, {\hat{𝐞}}_{3})$ , with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with index symbols such as i, j, k, used to identify an element of a set or array or sequence of variables).

When a unit vector in space is expressed, with Cartesian notation, as a linear combination of i, j, k, 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 coordinates

The three orthogonal unit vectors appropriate to cylindrical symmetry are:

$\hat{𝐬}$ (also designated $\hat{𝐞}$ or $\hat{𝝆}$ ), representing the direction along which the distance of the point from the axis of symmetry is measured;
$\hat{𝝋}$ , representing the direction of the motion that would be observed if the point were rotating counterclockwise about the symmetry axis;
$\hat{𝐳}$ , representing the direction of the symmetry axis;

They are related to the Cartesian basis $\hat{x}$ , $\hat{y}$ , $\hat{z}$ by:

\hat{𝐬}

=

\cos φ \hat{𝐱} + \sin φ \hat{𝐲}

\hat{𝝋}

=

- \sin φ \hat{𝐱} + \cos φ \hat{𝐲}

\hat{𝐳} = \hat{𝐳} .

It is important to note that $\hat{𝐬}$ and $\hat{𝝋}$ are functions of $φ$ , and are not constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. For a more complete description, see Jacobian matrix. The derivatives with respect to $φ$ are:

\frac{\partial \hat{𝐬}}{\partial φ} = - \sin φ \hat{𝐱} + \cos φ \hat{𝐲} = \hat{𝝋}

\frac{\partial \hat{𝝋}}{\partial φ} = - \cos φ \hat{𝐱} - \sin φ \hat{𝐲} = - \hat{𝐬}

\frac{\partial \hat{𝐳}}{\partial φ} = 𝟎 .

Spherical coordinates

The unit vectors appropriate to spherical symmetry are: $\hat{𝐫}$ , the direction in which the radial distance from the origin increases; $\hat{𝝋}$ , the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and $\hat{𝜽}$ , the direction in which the angle from the positive z axis is increasing. To minimize degeneracy, the polar angle is usually taken $0 \leq θ \leq 18 0^{\circ}$ . It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of $\hat{𝝋}$ and $\hat{𝜽}$ are often reversed. Here, the American "physics" convention^[1] is used. This leaves the azimuthal angle $φ$ defined the same as in cylindrical coordinates. The Cartesian relations are:

\hat{𝐫} = \sin θ \cos φ \hat{𝐱} + \sin θ \sin φ \hat{𝐲} + \cos θ \hat{𝐳}

\hat{𝜽} = \cos θ \cos φ \hat{𝐱} + \cos θ \sin φ \hat{𝐲} - \sin θ \hat{𝐳}

\hat{𝝋} = - \sin φ \hat{𝐱} + \cos φ \hat{𝐲}

The spherical unit vectors depend on both $φ$ and $θ$ , and hence there are 5 possible non-zero derivatives. For a more complete description, see Jacobian. The non-zero derivatives are:

\frac{\partial \hat{𝐫}}{\partial φ} = - \sin θ \sin φ \hat{𝐱} + \sin θ \cos φ \hat{𝐲} = \sin θ \hat{𝝋}

\frac{\partial \hat{𝐫}}{\partial θ} = \cos θ \cos φ \hat{𝐱} + \cos θ \sin φ \hat{𝐲} - \sin θ \hat{𝐳} = \hat{𝜽}

\frac{\partial \hat{𝜽}}{\partial φ} = - \cos θ \sin φ \hat{𝐱} + \cos θ \cos φ \hat{𝐲} = \cos θ \hat{𝝋}

\frac{\partial \hat{𝜽}}{\partial θ} = - \sin θ \cos φ \hat{𝐱} - \sin θ \sin φ \hat{𝐲} - \cos θ \hat{𝐳} = - \hat{𝐫}

\frac{\partial \hat{𝝋}}{\partial φ} = - \cos φ \hat{𝐱} - \sin φ \hat{𝐲} = - \sin θ \hat{𝐫} - \cos θ \hat{𝜽}

General unit vectors

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Common general themes of unit vectors occur throughout physics and geometry:^[2]

Unit vector	Nomenclature	Diagram
Tangent vector to a curve/flux line	$\hat{𝐭}$	A normal vector $\hat{𝐧}$ to the plane containing and defined by the radial position vector $r \hat{𝐫}$ and angular tangential direction of rotation $θ \hat{𝜽}$ 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	$\hat{𝐧}$ In terms of polar coordinates; $\hat{𝐧} = \hat{𝐫} \times \hat{𝜽}$
Binormal vector to tangent and normal	$\hat{𝐛} = \hat{𝐭} \times \hat{𝐧}$ ^[3]
Parallel to some axis/line	${\hat{𝐞}}_{∥}$	One unit vector ${\hat{𝐞}}_{∥}$ aligned parallel to a principle direction (red line), and a perpendicular unit vector ${\hat{𝐞}}_{⊥}$ is in any radial direction relative to the principle line.
Perpendicular to some axis/line in some radial direction	${\hat{𝐞}}_{⊥}$
Possible angular deviation relative to some axis/line	${\hat{𝐞}}_{∠}$	Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principle direction.

Curvilinear coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors ${\hat{𝐞}}_{n}$ equal to the degrees of freedom of the space. For ordinary 3-space, these vectors may be denoted ${\hat{𝐞}}_{1}, {\hat{𝐞}}_{2}, {\hat{𝐞}}_{3}$ . It is nearly always convenient to define the system to be orthonormal and right-handed:

${\hat{𝐞}}_{i} \cdot {\hat{𝐞}}_{j} = δ_{i j}$

${\hat{𝐞}}_{i} \cdot ({\hat{𝐞}}_{j} \times {\hat{𝐞}}_{k}) = ε_{i j k}$

where δ_ij is the Kronecker delta (which is one for i = j and zero else) and $ε_{i j k}$ is the Levi-Civita symbol (which is one for permutations ordered as ijk and minus one for permutations ordered as kji).

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

↑ Tevian Dray and Corinne A. Manogue,Spherical Coordinates, College Math Journal 34, 168-169 (2003).
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[1] Tevian Dray and Corinne A. Manogue,Spherical Coordinates, College Math Journal 34, 168-169 (2003).

[2] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[3] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[1]

[2]

[3]

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+In [[mathematics]], a '''unit vector''' in a [[normed vector space]] is a [[Vector space|vector]] (often a [[vector (geometry)|spatial vector]]) whose [[Norm (mathematics)|length]] is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a "[[circumflex|hat]]", like this: <math alt= i-hat>{\hat{\imath}}</math> (pronounced "i-hat").
+The '''normalized vector''' or '''[[Versor#Definition in linear algebra, geometry, and physics|versor]]''' <math alt= u-hat>\mathbf{\hat{u}}</math> of a non-zero vector '''u''' is the unit vector codirectional with '''u''', i.e.,
+:<math alt= "u-hat equals the vector u divided by its length">\mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}</math>
+where ||'''u'''|| is the [[Norm (mathematics)|norm]] (or length) of '''u'''.  The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''.
+The elements of a [[basis (linear algebra)|basis]] are usually chosen to be unit vectors. Every vector in the space may be written as a linear combination of unit vectors. The most commonly encountered bases are [[Cartesian coordinate system|Cartesian]], [[Polar coordinate system|polar]], and [[Spherical coordinate system|spherical]] coordinates. Each uses different unit vectors according to the symmetry of the coordinate system. Since these systems are encountered in so many different contexts, it is not uncommon to encounter different naming conventions than those used here.
+By definition, in [[Euclidean space]] the [[dot product]] of two unit vectors is simply the [[cosine]] of the angle between them. In three-dimensional Euclidean space, the [[cross product]] of two orthogonal unit vectors is another unit vector, orthogonal to both of them.
+==Orthogonal coordinates==
+===Cartesian coordinates===
+{{Main|Standard basis|Versor (physics)}}
+Unit vectors may be used to represent the axes of a [[Cartesian coordinate system]]. For instance, the unit vectors codirectional with the ''x'', ''y'', and ''z'' axes of a 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{\imath}} = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{\jmath}} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\,  \mathbf{\hat{k}} = \begin{bmatrix}0\\0\\1\end{bmatrix}</math>
+They are sometimes referred to as the [[Versor (physics)|versors]] of the coordinate system, and they form a set of mutually [[orthogonal]] unit vectors which, in the context of [[linear algebra]], is typically referred to as a [[standard basis]].
+They are often denoted using normal vector notation (e.g., '''''i''''' or <math alt= "vector i">\vec{\imath}</math>) rather than standard unit vector notation (e.g., <math alt= "unit vector i">\mathbf{\hat{\imath}}</math>). In most contexts it can be assumed that '''i''', '''j''', and '''k''', (or <math alt="vector i">\vec{\imath},</math> <math alt= "vector j">\vec{\jmath},</math> and <math alt= "vector k"> \vec{k}</math>) are versors of a 3-D Cartesian coordinate system. The notations <math alt="x-hat, y-hat, z-hat">(\mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\hat{z}})</math>, <math alt="x-hat sub 1, x-hat sub 2, x-hat sub 3">(\mathbf{\hat{x}}_1, \mathbf{\hat{x}}_2, \mathbf{\hat{x}}_3)</math>, <math alt="e-hat sub x, e-hat sub y, e-hat sub z">(\mathbf{\hat{e}}_x, \mathbf{\hat{e}}_y, \mathbf{\hat{e}}_z)</math>, or <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>, with or without [[Circumflex#Mathematics|hat]], are also used, 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'', used to identify an element of a set or array or sequence of variables).
+When a unit vector in space is expressed, with [[Cartesian coordinate system#Representing a vector with Cartesian notation|Cartesian notation]], as a linear combination of '''i''', '''j''', '''k''', 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===
+The three [[orthogonal]] unit vectors appropriate to cylindrical symmetry are:
+* <math alt="s-hat">\mathbf{\hat{s}}</math> (also designated <math alt="r-hat">\mathbf{\hat{e}}</math> or <math alt="rho-hat">\boldsymbol{\hat \rho}</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="s-hat">\mathbf{\hat{s}}</math> = <math alt="cosine of phi in the x-hat direction plus sine of phi in the y-hat direction">\cos \varphi\mathbf{\hat{x}} + \sin \varphi\mathbf{\hat{y}}</math>
+:<math alt="phi-hat">\boldsymbol{\hat \varphi}</math> = <math alt="minus the sine of phi in the x-hat direction plus the cosine of phi in the y-hat direction">-\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>
+It is important to note that <math alt="s-hat">\mathbf{\hat{s}}</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. For a more complete description, see [[Jacobian matrix]]. The derivatives with respect to <math>\varphi</math> are:
+:<math alt="partial derivative of s-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 \mathbf{\hat{s}}} {\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 s-hat">\frac{\partial \boldsymbol{\hat \varphi}} {\partial \varphi} = -\cos \varphi\mathbf{\hat{x}} - \sin \varphi\mathbf{\hat{y}} = -\mathbf{\hat{s}}</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 degeneracy, the polar angle is usually taken <math alt="zero is less than or equal to theta is less than or equal to 180 degrees">0\leq\theta\leq 180^\circ</math>. 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]]. 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 general themes of unit vectors occur throughout [[physics]] and [[geometry]]:<ref>{{cite book|title=Calculus (Schaum's Outlines Series)|edition=5th|publisher=Mc Graw Hill|author=F. Ayres, E. Mandelson|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|author=M. R. Spiegel, S. Lipschutz, 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 principle direction (red line), and a perpendicular unit vector <math> \mathbf{\hat{e}}_{\bot}\,\!</math> is in any radial direction relative to the principle 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 principle 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> 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 δ<sub>''ij''</sub> is the [[Kronecker delta]] (which is one for ''i'' = ''j'' and zero else) and  <math alt="epsilon sub i,j,k"> \varepsilon_{ijk} </math> is the [[Levi-Civita symbol]] (which is one for permutations ordered as ''ijk'' and minus one for permutations ordered as ''kji'').
+==See also==
+*[[Cartesian coordinate system]]
+*[[Coordinate system]]
+*[[Curvilinear coordinates]]
+*[[Four-velocity]]
+*[[Jacobian]]
+*[[Polar coordinate system]]
+*[[versor|Right versor]]
+*[[Unit interval]]
+* Unit [[unit square|square]], [[unit cube|cube]], [[unit circle|circle]], and [[unit sphere|sphere]]
+==References==
+{{Reflist}}
+*{{cite book|author=G. B. Arfken & H. J. Weber|title=Mathematical Methods for Physicists|edition=5th ed.|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 ed.|year=1998|publisher=McGraw-Hill|isbn=0-07-038203-4}}
+*{{cite book|first=David J.|last=Griffiths|title=Introduction to Electrodynamics|edition=3rd ed.|year=1998|publisher=Prentice Hall|isbn=0-13-805326-X}}
+{{DEFAULTSORT:Unit Vector}}
+[[Category:Linear algebra]]
+[[Category:Elementary mathematics]]
+[[Category:One]]
+[[Category:Vectors]]

Knot invariant: Difference between revisions

en>Hyacinth

Revision as of 02:42, 30 April 2013

Contents

Orthogonal coordinates

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

General unit vectors

Curvilinear coordinates

See also

References

Navigation menu

Knot invariant: Difference between revisions

en>Hyacinth

Revision as of 02:42, 30 April 2013

Orthogonal coordinates

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

General unit vectors

Curvilinear coordinates

See also

References

Navigation menu

Search