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). A unit vector is often denoted by a lowercase letter with a caret or “hat”, like this: $i-hat$ (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 $u-hat$ of a non-zero vector $u$ is the unit vector codirectional with $u$ , i.e.,

$u-hat equals the vector u divided by its length$

where $vector u between two pairs of vertical lines$ is the norm (or length) of $vector u$ . 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.

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

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.

$i-hat equals the 3 by 1 matrix 1,0,0; j-hat equals the 3 by 1 matrix 0,1,0; k-hat equals the 3 by 1 matrix 0,0,1$

These are often written using normal vector notation (e.g. i, or $vector i$ ) rather than the caret notation, and in most contexts it can be assumed that i, j, and k, (or $vector i$ $vector j$ and $vector k$ ) are versors of a Cartesian coordinate system (hence a tern of mutually orthogonal unit vectors). The notations $x-hat, y-hat, z-hat$ , $x-hat sub 1, x-hat sub 2, x-hat sub 3$ , $e-hat sub x, e-hat sub y, e-hat sub z$ , or $e-hat sub 1, e-hat sub 2, e-hat sub 3$ , 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). 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

The unit vectors appropriate to cylindrical symmetry are: $s-hat$ (also designated $r-hat$ or $rho-hat$ ), the distance from the axis of symmetry; $phi-hat$ , the angle measured counterclockwise from the positive x-axis; and $z-hat$ . They are related to the Cartesian basis $x-hat$ , $y-hat$ , $z-hat$ by:

$s-hat$ = $cosine of phi in the x-hat direction plus sine of phi in the y-hat direction$

$phi-hat$ = $minus the sine of phi in the x-hat direction plus the cosine of phi in the y-hat direction$

$z-hat equals z-hat$

It is important to note that $s-hat$ and $phi-hat$ are functions of $coordinate phi$ , 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 $\varphi$ are:

$partial derivative of s-hat with respect to phi equals minus sine of phi in the x-hat direction plus cosine of phi in the y-hat direction equals phi-hat$

$partial derivative of phi-hat with respect to phi equals minus cosine of phi in the x-hat direction minus sine of phi in the y-hat direction equals minus s-hat$

$partial derivative of z-hat with respect to phi equals zero$

Spherical coordinates

The unit vectors appropriate to spherical symmetry are: $r-hat$ , the radial distance from the origin; $phi-hat$ , the angle in the x-y plane counterclockwise from the positive x-axis; and $theta-hat$ , the angle from the positive z axis. To minimize degeneracy, the polar angle is usually taken $zero is less than or equal to theta is less than or equal to 180 degrees$ . It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of $phi-hat$ and $theta-hat$ 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:

$r-hat equals sin of theta times cosine of phi in the x-hat direction plus sine of theta times sine of phi in the y-hat direction plus cosine of theta in the z-hat direction$

$theta-hat equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction$

$phi-hat equals minus sine of phi in the x-hat direction plus cosine of phi in the y-hat direction$

The spherical unit vectors depend on both $phi$ and $theta$ , and hence there are 5 possible non-zero derivatives. For a more complete description, see Jacobian. The non-zero derivatives are:

$partial derivative of r-hat with respect to phi equals minus sine of theta times sine of phi in the x-hat direction plus sine of theta times cosine of phi in the y-hat direction equals sine of theta in the phi-hat direction$

$partial derivative of r-hat with respect to theta equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction equals theta-hat$

$partial derivative of theta-hat with respect to phi equals minus cosine of theta times sine of phi in the x-hat direction plus cosine of theta times cosine of phi in the y-hat direction equals cosine of theta in the phi-hat direction$

$partial derivative of theta-hat with respect to theta equals minus sine of theta times cosine of phi in the x-hat direction minus sine of theta times sine of phi in the y-hat direction minus cosine of theta in the z-hat direction equals minus r-hat$

$partial derivative of phi-hat with respect to phi equals minus cosine of phi in the x-hat direction minus sine of phi in the y-hat direction equals minus sine of theta in the r-hat direction minus cosine of theta in the theta-hat direction$

Curvilinear coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors $e-hat sub n$ equal to the degrees of freedom of the space. For ordinary 3-space, these vectors may be denoted $e-hat sub 1, e-hat sub 2, e-hat sub 3$ . It is nearly always convenient to define the system to be orthonormal and right-handed:

$e-hat sub i dot e-hat sub j equals Kronecker delta of i and j$

$e-hat sub i dot e-hat sub j cross e-hat sub k = epsilon sub ijk$

where δ_ij is the Kronecker delta (which is one for i = j and zero else) and $epsilon sub i,j,k$ is the Levi-Civita symbol (which is one for permutations ordered as ijk and minus one for permutations ordered as kji).

References

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.