define a unit vector
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a vector which has a magnitude of one.
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": {\displaystyle {\hat {\imath }}} {\hat {\imath }} (pronounced "i-hat"). The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.
The same construct is used to specify spatial directions in 3D. As illustrated, each unique direction is equivalent numerically to a point on the unit sphere.
Examples of two 2D direction vectors
Examples of two 3D direction vectors
The normalized vector or versor û of a non-zero vector u is the unit vector in the direction of u, i.e.,
{\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{|\mathbf {u} |}}} {\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{|\mathbf {u} |}}}
where |u| is the norm (or length) of u. The term normalized vector is sometimes used as a synonym for unit vector.
Unit vectors are often chosen to form the basis of a vector space. Every vector in the space may be written as a linear combination of unit vectors.
By definition, in a Euclidean space the dot product of two unit vectors is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the cross product of two arbitrary unit vectors is a third vector orthogonal to both of them having length equal to the sine of the smaller subtended angle. The normalized cross product corrects for this varying length, and yields the mutually orthogonal unit vector to the two inputs, applying the right-hand rule to resolve one of two possible directions.
Orthogonal coordinates Edit
Cartesian coordinates Edit
Main article: Standard basis
Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are
{\displaystyle \mathbf {\hat {i}} ={\begin{bmatrix}1\\0\\0\end{bmatrix}},\,\,\mathbf {\hat {j}} ={\begin{bmatrix}0\\1\\0\end{bmatrix}},\,\,\mathbf {\hat {k}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}} \mathbf {\hat {i}} ={\begin{bmatrix}1\\0\\0\end{bmatrix}},\,\,\mathbf {\hat {j}} ={\begin{bmatrix}0\\1\\0\end{bmatrix}},\,\,\mathbf {\hat {k}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}
They are sometimes referred to as the versors of the coordinate system, and they form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra.
They are often denoted using normal vector notation (e.g., i or {\displaystyle {\vec {\imath }}} {\vec {\imath }}) rather than standard unit vector notation (e.g., {\displaystyle \mathbf {\hat {\imath }} } \mathbf {\hat {\imath }} ). In most contexts it can be assumed that i, j, and k, (or {\displaystyle {\vec {\imath }},} {\vec {\imath }}, {\displaystyle {\vec {\jmath }},} {\vec {\jmath }}, and {\displaystyle {\vec {k}}} {\vec {k}}) are versors of a 3-D Cartesian coordinate system. The notations {\displaystyle (\mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} )} (\mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} ), {\displaystyle (\mathbf {\hat {x}} _{1},\mathbf {\hat {x}} _{2},\mathbf {\hat {x}} _{3})} (\mathbf {\hat {x}} _{1},\mathbf {\hat {x}} _{2},\mathbf {\hat {x}} _{3}), {\displaystyle (\mathbf {\hat {e}} _{x},\mathbf {\hat {e}} _{y},\mathbf {\hat {e}} _{z})} (\mathbf {\hat {e}} _{x},\mathbf {\hat {e}} _{y},\mathbf {\hat {e}} _{z}), or {\displaystyle (\mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3})} (\mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3}), with or without hat, 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).
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).