Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length A unit vector is often denoted by a lowercase letter with a superscribed caret or “hat”, like this: ${\hat{\imath}}$ (pronounced "i-hat"). Caret is the name for the symbol ^ in ASCII and some other Character sets Its Unicode code point is U+005E and its ASCII code in hexadecimal is 5E

In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R This follows from the formula for the dot product, since the lengths are both 1.

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

$\boldsymbol{\hat{u}} = \frac{\boldsymbol{u}}{\|\boldsymbol{u}\|}.$

where $\|\boldsymbol{u}\|$ is the norm (or length) of $\boldsymbol{u}$ . In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length 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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial 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.

1 Cartesian coordinates
2 Cylindrical coordinates
3 Spherical coordinates
4 Curvilinear Coordinates
5 References
6 See also

Cartesian coordinates

In the three-dimensional Cartesian coordinate system, the unit vectors codirectional with the x, y, and z axes are sometimes referred to as versors of the coordinate system. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, a versor is a directed great-circle arc that corresponds to a Quaternion of norm one

$\mathbf{\hat{\boldsymbol{\imath}}} = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{\boldsymbol{\jmath}}} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\, \mathbf{\hat{\boldsymbol{k}}} = \begin{bmatrix}0\\0\\1\end{bmatrix}$

These are often written using normal vector notation (e. g. i, or $\vec{\imath}$ ) rather than the caret notation, and in most contexts it can be assumed that i, j, and k, (or $\vec{\imath}, \vec{\jmath},$ and $\vec{k}$ ) are versors of a Cartesian coordinate system (hence a tern of reciprocally orthogonal unit vectors). In Mathematics, two Vectors are orthogonal if they are Perpendicular, i The notations $(\boldsymbol\hat{x}, \boldsymbol\hat{y}, \boldsymbol\hat{z})$ , $(\boldsymbol\hat{x}_1, \boldsymbol\hat{x}_2, \boldsymbol\hat{x}_3)$ , $(\boldsymbol\hat{e}_x, \boldsymbol\hat{e}_y, \boldsymbol\hat{e}_z)$ , or $(\boldsymbol\hat{e}_1, \boldsymbol\hat{e}_2, \boldsymbol\hat{e}_3)$ , with or without hat/caret, 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). The word index is used in variety of senses in Mathematics. In perhaps the most frequent sense an index is a Superscript These vectors represent an example of standard basis. In Mathematics, the standard basis (also called natural basis or canonical basis) of the n- dimensional Euclidean space

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". In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 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 unit vectors appropriate to cylindrical symmetry are: $\boldsymbol{\hat{s}}$ (also designated $\boldsymbol{\hat{r}}$ or $\boldsymbol{\hat \rho}$ ), the distance from the axis of symmetry; $\boldsymbol{\hat \phi}$ , the angle measured counterclockwise from the positive x-axis; and $\boldsymbol{\hat{z}}$ . They are related to the Cartesian basis $\hat{x}, \hat{y}, \hat{z}$ by:

$\boldsymbol{\hat{s}}$ = $\cos \phi\boldsymbol{\hat{x}} + \sin \phi\boldsymbol{\hat{y}}$

$\boldsymbol{\hat \phi}$ = $-\sin \phi\boldsymbol{\hat{x}} + \cos \phi\boldsymbol{\hat{y}}$

$\boldsymbol{\hat{z}}=\boldsymbol{\hat{z}}.$

It is important to note that $\boldsymbol{\hat{s}}$ and $\boldsymbol{\hat \phi}$ 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. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. The derivatives with respect to $φ$ are:

$\frac{\partial \boldsymbol{\hat{s}}} {\partial \phi} = -\sin \phi\boldsymbol{\hat{x}} + \cos \phi\boldsymbol{\hat{y}} = \boldsymbol{\hat \phi}$

$\frac{\partial \boldsymbol{\hat \phi}} {\partial \phi} = -\cos \phi\boldsymbol{\hat{x}} - \sin \phi\boldsymbol{\hat{y}} = -\boldsymbol{\hat{s}}$

$\frac{\partial \boldsymbol{\hat{z}}} {\partial \phi} = \mathbf{0}.$

Spherical coordinates

The unit vectors appropriate to spherical symmetry are: $\boldsymbol{\hat{r}}$ , the radial distance from the origin; $\boldsymbol{\hat{\phi}}$ , the angle in the x-y plane counterclockwise from the positive x-axis; and $\boldsymbol{\hat \theta}$ , the angle from the positive z axis. To minimize degeneracy, the polar angle is usually taken $0\leq\theta\leq 180^\circ$ . It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of $\boldsymbol{\hat \phi}$ and $\boldsymbol{\hat \theta}$ are often reversed. Here, the American naming convention is used. This leaves the azimuthal angle $φ$ defined the same as in cylindrical coordinates. The Cartesian relations are:

$\boldsymbol{\hat{r}} = \sin \theta \cos \phi\boldsymbol{\hat{x}} + \sin \theta \sin \phi\boldsymbol{\hat{y}} + \cos \theta\boldsymbol{\hat{z}}$

$\boldsymbol{\hat \theta} = \cos \theta \cos \phi\boldsymbol{\hat{x}} + \cos \theta \sin \phi\boldsymbol{\hat{y}} - \sin \theta\boldsymbol{\hat{z}}$

$\boldsymbol{\hat \phi} = - \sin \phi\boldsymbol{\hat{x}} + \cos \phi\boldsymbol{\hat{y}}$

The spherical unit vectors depend on both $φ$ and $θ$ , and hence there are 5 possible non-zero derivates. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane For a more complete description, see Jacobian. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. The non-zero derivatives are:

$\frac{\partial \boldsymbol{\hat{r}}} {\partial \phi} = -\sin \theta \sin \phi\boldsymbol{\hat{x}} + \sin \theta \cos \phi\boldsymbol{\hat{y}} = \sin \theta\boldsymbol{\hat \phi}$

$\frac{\partial \boldsymbol{\hat{r}}} {\partial \theta} =\cos \theta \cos \phi\boldsymbol{\hat{x}} + \cos \theta \sin \phi\boldsymbol{\hat{y}} - \sin \theta\boldsymbol{\hat{z}}= \boldsymbol{\hat \theta}$

$\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \phi} =-\cos \theta \sin \phi\boldsymbol{\hat{x}} + \cos \theta \cos \phi\boldsymbol{\hat{y}} = \cos \theta\boldsymbol{\hat \phi}$

$\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \theta} = -\sin \theta \cos \phi\boldsymbol{\hat{x}} - \sin \theta \sin \phi\boldsymbol{\hat{y}} - \cos \theta\boldsymbol{\hat{z}} = -\boldsymbol{\hat{r}}$

$\frac{\partial \boldsymbol{\hat{\phi}}} {\partial \phi} = -\cos \phi\boldsymbol{\hat{x}} - \sin \phi\boldsymbol{\hat{y}} = -\cos \theta\boldsymbol{\hat{\theta}} - \sin \theta\boldsymbol{\hat{r}}$

Curvilinear Coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors $\boldsymbol\hat{e}_n$ equal to the degrees of freedom of the space. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors For ordinary 3-space, these vectors may be denoted $\boldsymbol{\hat{e}_1}, \boldsymbol{\hat{e}_2}, \boldsymbol{\hat{e}_3}$ . It is nearly always convenient to define the system to be orthonormal and right-handed:

$\boldsymbol{\hat{e}_i} \cdot \boldsymbol{\hat{e}_j} = \delta_{ij}$

$\boldsymbol{\hat{e}_1} \cdot (\boldsymbol{\hat{e}_2} \times \boldsymbol{\hat{e}_3}) = 1$

where δ_ij is the Kronecker delta. For the related yet different principle relating to electromagnetic coils see Right hand grip rule. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

References

G. B. Arfken & H. J. Weber (2000). Mathematical Methods for Physicists, 5th ed. , Academic Press. ISBN 0-12-059825-6.
Spiegel, Murray R. (1998). Schaum's Outlines: Mathematical Handbook of Formulas and Tables, 2nd ed. , McGraw-Hill. ISBN 0-07-038203-4.
Griffiths, David J. (1998). Introduction to Electrodynamics, 3rd ed. , Prentice Hall. ISBN 0-13-805326-X.

Dictionary

-noun

(vector algebra) A vector with length 1.

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