Unit Vector

Article Index for
Unit

Website Links For
Unit

Information About

Unit Vector

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. The normalized vector '''û''' of a non-zero vector '''u''' is the unit vector codirectional with '''u''', i.e.,
	Where '''u''' Is The	"http://wwwinformationdelightinfo/encyclopedia/entry/Norm_(mathematics)" class="copylinks">Norm (or length) of '''u''' The term ''normalized vector'' is sometimes used simply as a synonym for ''unit vector''

These are not always written with a hat/caret - sometimes they are written using normal vector notations. However, it can generally be assumed that

ec{i}, ec{j}, ec{k}

are unit vectors in most contexts. When dealing with electromagnetics topics, some people prefer to use

\hat{x}, \hat{y}, \hat{z}

for the 3 unit vectors on the 3 axes. Sometimes, these unit vectors are also written as

ec{e_1}, ec{e_2}, ec{e_3}

respectively.

Other coordinate systems, such as Polar Coordinates or Spherical Coordinates use different unit vectors; notations vary.

CYLINDRICAL COORDINATES

In areas of physics such as electrodynamics, a cylindrical coordinate system may be more appropriate if a problem has such symmetry. The corresponding unit vectors

\boldsymbol{\hat{s}}, \boldsymbol{\hat \phi}, \boldsymbol{\hat{z}}

relate 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 ''not'' constant in direction (they are functions of

\phi

). When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. Hence the derivatives with respect to

\phi

are

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

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

rac{\partial \boldsymbol{\hat{z}}} {\partial \phi} = 0

SPHERICAL COORDINATES

Spherical coordinates are appropriate for problems involving spherical symmetry. The unit vectors

\boldsymbol{\hat{r}}, \boldsymbol{\hat \phi}, \boldsymbol{\hat 	heta}

are commonly used in most physics textbooks, with

0\leq	heta\leq 180^\circ

and Φ the same as in cylindrical coordinates. The Cartesian relations are

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

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

heta

, and hence there are 5 possible derivates.

rac{\partial \boldsymbol{\hat{r}}} {\partial \phi} = -\sin 	heta \sin \phi\boldsymbol{\hat{x}} + \sin 	heta \cos \phi\boldsymbol{\hat{y}} = \sin 	heta\boldsymbol{\hat \phi}

rac{\partial \boldsymbol{\hat{r}}} {\partial 	heta} =\cos 	heta \cos \phi\boldsymbol{\hat{x}} + \cos 	heta \sin \phi\boldsymbol{\hat{y}} - \sin 	heta\boldsymbol{\hat{z}}= \boldsymbol{\hat 	heta}

rac{\partial \boldsymbol{\hat{	heta}}} {\partial \phi} =-\cos 	heta \sin \phi\boldsymbol{\hat{x}} + \cos 	heta \cos \phi\boldsymbol{\hat{y}} = \cos 	heta\boldsymbol{\hat \phi}

rac{\partial \boldsymbol{\hat{	heta}}} {\partial 	heta} = -\sin 	heta \cos \phi\boldsymbol{\hat{x}} - \sin 	heta \sin \phi\boldsymbol{\hat{y}} - \cos 	heta\boldsymbol{\hat{z}} = -\boldsymbol{\hat{r}}

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

The derivatives for both the spherical and cylindrical unit vectors must be calculated when converting differential formulas, such as the Laplacian and Divergence , to the respective coordinate system.