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Example:Finding Unit Vector

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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 "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).




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.