Editing Unit vector (section)

===Cartesian coordinates===
{{Main|Standard basis}}

Unit vectors may be used to represent the axes of a [[Cartesian coordinate system]]. For instance, the standard ''unit vectors'' in the direction of the ''x'', ''y'', and ''z'' axes of a [[Cartesian coordinate system|three dimensional Cartesian coordinate system]] are

:<math alt="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">
\mathbf{\hat{x}} = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{y}} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\,  \mathbf{\hat{z}} = \begin{bmatrix}0\\0\\1\end{bmatrix}</math>

They form a set of mutually [[orthogonal]] ''unit vectors'', typically referred to as a [[standard basis]] in [[linear algebra]].

They are often denoted using common [[vector notation]] (e.g., '''x''' or <math alt="vector i">\vec{x}</math>) rather than standard unit vector notation (e.g., '''x̂'''). In most contexts it can be assumed that '''x''', '''y''', and '''z''', (or <math alt="vector i">\vec{x},</math> <math alt="vector j">\vec{y},</math> and <math alt="vector k"> \vec{z}</math>) are versors of a 3-D [[Cartesian coordinate system]]. The notations ('''î''', '''ĵ''', '''k̂'''), ('''x̂<sub>1</sub>''', '''x̂<sub>2</sub>''', '''x̂<sub>3</sub>'''), ('''ê<sub>x</sub>''', '''ê<sub>y</sub>''', '''ê<sub>z</sub>'''), or ('''ê<sub>1</sub>''', '''ê<sub>2</sub>''', '''ê<sub>3</sub>'''), with or without [[Hat notation|hat]], are also used,<ref name=":0" /> particularly in contexts where '''i''', '''j''', '''k''' might lead to confusion with another quantity (for instance with [[Indexed family|index]] symbols such as ''i'', ''j'', ''k'', which are used to identify an element of a set or array or sequence of variables).

When a unit vector in space is expressed in [[Cartesian coordinate system#Representing a vector with Cartesian notation|Cartesian notation]] as a linear combination of '''x''', '''y''', '''z''', 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 (mathematics)|orientation]] (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis ([[vector (geometry)|vector]]).