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: (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 of a non-zero vector is the unit vector codirectional with, i.e.,
where is the norm (or length) of . The term normalized vector is sometimes used as a
synonym for unit vector. (Our solved example in
mathguru.com uses this concept)
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.
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.
These are often written using normal vector notation (e.g. i, or ) rather
than the caret notation, and in most contexts it can be assumed that i, j, and k, (or and ) are versors of a Cartesian coordinate system
(hence a tern of mutually orthogonal unit vectors). The notations , , , or , 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).
http://en.wikipedia.org/wiki/Unit_vector
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