http://en.wikipedia.org/wiki/Normalized_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). A unit vector is often denoted by a lowercase letter with a "hat", like this: http://upload.wikimedia.org/math/a/b/6/ab6d5202f926ed23242a4754abf0a2c9.pngvector(Normalized vector)" /> (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 http://upload.wikimedia.org/math/7/5/e/75ee590ab63d3067105f760851f7918b.pngvector(Normalized vector)" /> of a non-zero vector u is the unit vector codirectional with u, i.e.,

http://upload.wikimedia.org/math/1/6/0/160cba30d26b18903acfd568fe2a89e9.pngvector(Normalized vector)" />

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 areCartesian, 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.

Orthogonal coordinates[edit]

Cartesian coordinates[edit]

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.

http://upload.wikimedia.org/math/8/a/1/8a1dd5d83a20979bddbf5672ed65c022.pngvector(Normalized vector)" />

These are often written using normal vector notation (e.g. i, or http://upload.wikimedia.org/math/0/7/1/071f9f13e1a69c07e0d5af2381be5226.pngvector(Normalized vector)" />) rather than the circumflex notation, and in most contexts it can be assumed that i, j, and k, (or http://upload.wikimedia.org/math/0/c/e/0ce3c8c83ac88a97da62f18e619d599c.pngvector(Normalized vector)" />) are versors of a Cartesian coordinate system (hence a term of mutually orthogonal unit vectors). The notations http://upload.wikimedia.org/math/7/c/3/7c32a6f9cdd3dcf942d4c27efc3db664.pngvector(Normalized vector)" />, or http://upload.wikimedia.org/math/0/4/c/04cdc038d644837d4b3012533f973c64.pngvector(Normalized vector)" />, with or without hat/circumflex, 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 a 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[edit]

The unit vectors appropriate to cylindrical symmetry are: http://upload.wikimedia.org/math/9/5/7/9579abb6959a2042af3ce54a68651343.pngvector(Normalized vector)" /> (also designated http://upload.wikimedia.org/math/1/3/e/13e62c9c781fb977fb4895c5bc6084a0.pngvector(Normalized vector)" />), the distance from the axis of symmetry; http://upload.wikimedia.org/math/2/1/a/21adc7e05a467fdb0ec5d99d39845db9.pngvector(Normalized vector)" />, the angle measured counterclockwise from the positive x-axis; and http://upload.wikimedia.org/math/6/2/2/622fb40afbb1f9fcfbf7ff9c98f11a1b.pngvector(Normalized vector)" />. They are related to the Cartesian basis http://upload.wikimedia.org/math/8/7/5/875326710761d5ed42bdc6e30c4bf962.pngvector(Normalized vector)" /> by:

http://upload.wikimedia.org/math/2/c/0/2c08351360892cf241d630875a66f920.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/f/0/0/f0099a5079cadea4f9aa4a0da9e9a84f.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/0/b/1/0b10e5ae56e96423bc10a50986a72747.pngvector(Normalized vector)" />

It is important to note that http://upload.wikimedia.org/math/9/5/7/9579abb6959a2042af3ce54a68651343.pngvector(Normalized vector)" /> and http://upload.wikimedia.org/math/2/1/a/21adc7e05a467fdb0ec5d99d39845db9.pngvector(Normalized vector)" /> are functions of http://upload.wikimedia.org/math/3/5/3/3538eb9c84efdcbd130c4c953781cfdb.pngvector(Normalized vector)" />, 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 http://upload.wikimedia.org/math/3/5/3/3538eb9c84efdcbd130c4c953781cfdb.pngvector(Normalized vector)" /> are:

http://upload.wikimedia.org/math/9/c/7/9c7d08e974b689be895b14e35f7a2e14.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/d/c/2/dc2d92865693b663aa607467d3dbfc30.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/9/8/e/98e7eab3cae485f65c17ac696d3bb300.pngvector(Normalized vector)" />

Spherical coordinates[edit]

The unit vectors appropriate to spherical symmetry are: http://upload.wikimedia.org/math/b/c/9/bc9a431c46a5c00d1371e3e33c08e51b.pngvector(Normalized vector)" />, the direction in which the radial distance from the origin increases; http://upload.wikimedia.org/math/7/c/d/7cdb2c1cc258befce2b93d1fe5f93c28.pngvector(Normalized vector)" />, the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and http://upload.wikimedia.org/math/2/d/7/2d7b1378aa0b6d0b2f95607c2f4e8194.pngvector(Normalized vector)" />, the direction in which the angle from the positive z axis is increasing. To minimize degeneracy, the polar angle is usually taken http://upload.wikimedia.org/math/b/3/e/b3e81b316866519378bec2c2431c248d.pngvector(Normalized vector)" />. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of http://upload.wikimedia.org/math/2/1/a/21adc7e05a467fdb0ec5d99d39845db9.pngvector(Normalized vector)" /> and http://upload.wikimedia.org/math/2/d/7/2d7b1378aa0b6d0b2f95607c2f4e8194.pngvector(Normalized vector)" /> are often reversed. Here, the American "physics" convention^[1] is used. This leaves the azimuthal angle http://upload.wikimedia.org/math/3/5/3/3538eb9c84efdcbd130c4c953781cfdb.pngvector(Normalized vector)" /> defined the same as in cylindrical coordinates. The Cartesian relations are:

http://upload.wikimedia.org/math/a/1/a/a1a303d2623dcaa002809188599dc9c7.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/8/1/9/8197fb6a05ccaf7b624a4a6fe24c5567.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/7/9/e/79eb09cbb46123591264cf57618a757b.pngvector(Normalized vector)" />

The spherical unit vectors depend on both http://upload.wikimedia.org/math/3/5/3/3538eb9c84efdcbd130c4c953781cfdb.pngvector(Normalized vector)" /> and , and hence there are 5 possible non-zero derivatives. For a more complete description, see Jacobian. The non-zero derivatives are:

http://upload.wikimedia.org/math/e/4/e/e4ee7f9f099d2bba049b20d7c794d6a6.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/3/d/a/3da7414c7c0bb487636c16ad35afc83d.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/6/e/5/6e5c799ab8cb25fa75808bcef0849cc2.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/a/6/d/a6d17ab06af0e793b07868bbb31d83f8.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/6/0/a/60a35fda3ae1ca9bd78bdeea2f09a28c.pngvector(Normalized vector)" />

General unit vectors[edit]

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

Curvilinear coordinates[edit]

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors http://upload.wikimedia.org/math/f/8/8/f882eae30b22c9febd7918e29f705da8.pngvector(Normalized vector)" /> equal to the degrees of freedom of the space. For ordinary 3-space, these vectors may be denoted http://upload.wikimedia.org/math/8/1/a/81ad21835de2ed2c474c409bdc159f11.pngvector(Normalized vector)" />. It is nearly always convenient to define the system to be orthonormal and right-handed:

http://upload.wikimedia.org/math/9/4/2/942656a133049621065c31e269ebf64d.pngvector(Normalized vector)" />

http://upload.wikimedia.org/math/2/1/7/217be02286de60c0d022b29dc04fb618.pngvector(Normalized vector)" />

where δ_ij is the Kronecker delta (which is one for i = j and zero else) and http://upload.wikimedia.org/math/8/3/a/83a43d12ddc5e0ebdb49d3d3a28e41f3.pngvector(Normalized vector)" /> is the Levi-Civita symbol (which is one for permutations ordered as ijk and minus one for permutations ordered askji).

See also[edit]

• Cartesian coordinate system
• Coordinate system
• Curvilinear coordinates
• Four-velocity
• Jacobian
• Polar coordinate system
• Right versor
• Unit interval
• Unit square, cube, circle, and sphere

References[edit]

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