UNIT VECTOR

(Redirected from Unit vectors)
In mathematics, a 'unit vector' in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1 (the unit length). A unit vector is often written with a superscribed caret or “hat”, like this

{hat{imath}}

(pronounced "i-hat").
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' or 'versor'

oldsymbol{hat{u}}

of a non-zero vector

oldsymbol{u}

is the unit vector codirectional with

oldsymbol{u}

, i.e.,
:

oldsymbol{hat{u}} = rac{oldsymbol{u}}{|oldsymbol{u}|}.

where

|oldsymbol{u}|

is the norm (or length) of

oldsymbol{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. Usually, a little context should enable the astute reader to substitute the names being used for those given here.

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

Curvilinear Coordinates

References

Cartesian coordinates

In the 3-Dimensional Cartesian coordinate system, the unit vectors codirectional with the ''x'', ''y'', and ''z'' axes are often referred to as 'versors' of the coordinate system and denoted '''i''', '''j''', and '''k''', respectively.
:

mathbf{hat{i}} = egin{bmatrix}1\0\0end{bmatrix}, ,, mathbf{hat{j}} = egin{bmatrix}0\1\0end{bmatrix}, ,,  mathbf{hat{k}} = egin{bmatrix}0\0\1end{bmatrix}

These are sometimes written using normal vector notation rather than the hat/caret notation, and it can generally be assumed that

ec{imath}, ec{jmath}, ec{k}

are versors in most contexts. The notations

(oldsymbolhat{x}, oldsymbolhat{y}, oldsymbolhat{z})

(oldsymbolhat{x}_1, oldsymbolhat{x}_2, oldsymbolhat{x}_3)

, or

(oldsymbolhat{e}_x, oldsymbolhat{e}_y, oldsymbolhat{e}_z)

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). These vectors represent an example of standard basis.
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 unit vectors appropriate to cylindrical symmetry are:

oldsymbol{hat{s}}

(also designated

oldsymbol{hat{r}}

oldsymbol{hat 
ho}

), the distance from the axis of symmetry;

oldsymbol{hat phi}

, the angle measured counterclockwise from the positive ''x''-axis; and

oldsymbol{hat{z}}

. They are related to the Cartesian basis

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

by:
:

oldsymbol{hat{s}}

cos phioldsymbol{hat{x}} + sin phioldsymbol{hat{y}}

oldsymbol{hat phi}

-sin phioldsymbol{hat{x}} + cos phioldsymbol{hat{y}}

oldsymbol{hat{z}}=oldsymbol{hat{z}}

It is important to note that

oldsymbol{hat{s}}

and

oldsymbol{hat phi}

are functions of

phi

, 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

phi

are:
:

rac{partial oldsymbol{hat{s}}} {partial phi} = -sin phioldsymbol{hat{x}} + cos phioldsymbol{hat{y}} = oldsymbol{hat phi}

rac{partial oldsymbol{hat phi}} {partial phi} = -cos phioldsymbol{hat{x}} - sin phioldsymbol{hat{y}} = -oldsymbol{hat{s}}

rac{partial oldsymbol{hat{z}}} {partial phi} = 0

Spherical coordinates

The unit vectors appropriate to spherical symmetry are:

oldsymbol{hat{r}}

, the radial distance from the origin;

oldsymbol{hat{phi}}

, the angle in the ''x''-''y'' plane counterclockwise from the positive ''x''-axis; and

oldsymbol{hat 	heta}

, the angle from the positive ''z'' axis. To minimize degeneracy, the polar angle is usually taken

0leq	hetaleq 180^circ

. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of

oldsymbol{hat phi}

and

oldsymbol{hat 	heta}

are often reversed. Here, the American naming convention is used. This leaves the azimuthal angle

phi

defined the same as in cylindrical coordinates. The Cartesian relations are:
:

oldsymbol{hat{r}} = sin 	heta cos phioldsymbol{hat{x}}  + sin 	heta sin phioldsymbol{hat{y}} + cos 	hetaoldsymbol{hat{z}}

oldsymbol{hat 	heta} = cos 	heta cos phioldsymbol{hat{x}} + cos 	heta sin phioldsymbol{hat{y}} - sin 	hetaoldsymbol{hat{z}}

oldsymbol{hat phi} = -sin phioldsymbol{hat{x}} + cos phioldsymbol{hat{y}}

The spherical unit vectors depend on both

phi

and

heta

, and hence there are 5 possible non-zero derivates. For a more complete description, see Jacobian. The non-zero derivatives are:
:

rac{partial oldsymbol{hat{r}}} {partial phi} = -sin 	heta sin phioldsymbol{hat{x}} + sin 	heta cos phioldsymbol{hat{y}} = sin 	hetaoldsymbol{hat phi}

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

rac{partial oldsymbol{hat{	heta}}} {partial phi} =-cos 	heta sin phioldsymbol{hat{x}} + cos 	heta cos phioldsymbol{hat{y}} = cos 	hetaoldsymbol{hat phi}

rac{partial oldsymbol{hat{	heta}}} {partial 	heta} = -sin 	heta cos phioldsymbol{hat{x}} - sin 	heta sin phioldsymbol{hat{y}} - cos 	hetaoldsymbol{hat{z}} = -oldsymbol{hat{r}}

rac{partial oldsymbol{hat{phi}}} {partial phi} = -cos phioldsymbol{hat{x}} - sin phioldsymbol{hat{y}} = -cos 	hetaoldsymbol{hat{	heta}} - sin 	hetaoldsymbol{hat{r}}

Curvilinear Coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors

oldsymbolhat{e}_n

equal to the degrees of freedom of the space. For ordinary 3-space, these vectors may be denoted

oldsymbol{hat{e}_1}, oldsymbol{hat{e}_2}, oldsymbol{hat{e}_3}

. It is nearly always convenient to define the system to be orthonormal and right-handed:

oldsymbol{hat{e}_i} cdot oldsymbol{hat{e}_j} = delta_{ij}

oldsymbol{hat{e}_1} cdot (oldsymbol{hat{e}_2} 	imes oldsymbol{hat{e}_3}) = 1

where δ_''ij'' is the Kronecker delta.

References

★ Mathematical Methods for Physicists, G. B. Arfken & H. J. Weber, , , Academic Press, 2000, ISBN 0-12-059825-6

★ Schaum's Outlines: Mathematical Handbook of Formulas and Tables, , Murray R., Spiegel, McGraw-Hill, 1998, ISBN 0-07-038203-4

★ Introduction to Electrodynamics, , David J., Griffiths, Prentice Hall, 1998, ISBN 0-13-805326-X

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UNIT VECTOR

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

Curvilinear Coordinates

References

See also

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