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Encyclopedia > Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. A unit vector is often written with a superscribed caret or “hat”, like this {hat{imath}} (pronounced "i-hat"). For other meanings of mathematics or math, see mathematics (disambiguation). ... 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 any real vector space Rn. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... This article is about vectors that have a particular relation to the spatial coordinates. ...


In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1. Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...


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

mathbf{hat{u}} = frac{mathbf{u}}{|mathbf{u}|}.

where ||u|| is the norm (or length) of u. The term normalized vector is sometimes used as a synonym for unit vector. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero 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, with the components of each being given by direction cosines. 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. Usually, a little context should enable the astute reader to substitute the names being used for those given here. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... Fig. ... The polar coordinate system is a two-dimensional coordinate system in which points are given by an angle and a distance from the pole, called the origin in the Cartesian coordinate system. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

Contents

Cartesian coordinates

In the 3-Dimensional Cartesian coordinate system, the unit vectors along the x, y, and z axes are usually denoted i, j, and k, respectively. Fig. ...

mathbf{hat{i}} = begin{bmatrix}100end{bmatrix}, ,, mathbf{hat{j}} = begin{bmatrix}010end{bmatrix}, ,, mathbf{hat{k}} = begin{bmatrix}001end{bmatrix}

These are sometimes written using normal vector notation rather than the hat/caret notation, and it can generally be assumed that vec{i}, vec{j}, vec{k} are unit vectors in most contexts. The notations (boldsymbolhat{x}, boldsymbolhat{y}, boldsymbolhat{z}), (boldsymbolhat{x}_1, boldsymbolhat{x}_2, boldsymbolhat{x}_3), or (boldsymbolhat{e}_x, boldsymbolhat{e}_y, boldsymbolhat{e}_z) are also used, particularly in contexts where i, j, k might lead to confusion with another quantity.


Cylindrical coordinates

The unit vectors appropriate to cylindrical symmetry are: boldsymbol{hat{s}}, 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 phiboldsymbol{hat{x}} + sin phiboldsymbol{hat{y}}
boldsymbol{hat phi} = -sin phiboldsymbol{hat{x}} + cos phiboldsymbol{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. The derivatives with respect to φ are: In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

frac{partial boldsymbol{hat{s}}} {partial phi} = -sin phiboldsymbol{hat{x}} + cos phiboldsymbol{hat{y}} = boldsymbol{hat phi}
frac{partial boldsymbol{hat phi}} {partial phi} = -cos phiboldsymbol{hat{x}} - sin phiboldsymbol{hat{y}} = -boldsymbol{hat{s}}
frac{partial boldsymbol{hat{z}}} {partial phi} = 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 0leqthetaleq 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: Fig. ...

boldsymbol{hat{r}} = sin theta cos phiboldsymbol{hat{x}} + sin theta sin phiboldsymbol{hat{y}} + cos thetaboldsymbol{hat{z}}
boldsymbol{hat theta} = cos theta cos phiboldsymbol{hat{x}} + cos theta sin phiboldsymbol{hat{y}} - sin thetaboldsymbol{hat{z}}
boldsymbol{hat phi} = -sin phiboldsymbol{hat{x}} + cos phiboldsymbol{hat{y}}

The spherical unit vectors depend on both φ and θ, and hence there are 5 possible non-zero derivates. For a more complete description, see Jacobian. The non-zero derivatives are: In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

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

Curvilinear Coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors boldsymbolhat{e}_n equal to the degrees of freedom of the space. 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: 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 in the collection. ... The left-handed orientation is shown on the left, and the right-handed on the right. ...


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 δij is the Kronecker delta. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...


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.

See also


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