Unit length

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: {\hat{\imath}} (pronounced "i-hat").

The normalized vector or versor \bold{\hat{u}} of a non-zero vector u is the unit vector codirectional with u, i.e.,

\bold{\hat{u}} = \frac{\bold{u}}{\|\bold{u}\|}

where ||u|| is the norm (or length) of 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.

By definition, in Euclidean space the dot product of two unit vectors is simply the cosine of the angle between them. In three-dimensional Euclidean space, the cross product of two orthogonal unit vectors is another unit vector, orthogonal to both of them.

Orthogonal coordinates

Cartesian coordinates

Main articles: Standard basis and Versor (physics)

Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the unit vectors codirectional with the x, y, and z axes of a three dimensional Cartesian coordinate system are

\mathbf{\hat{\boldsymbol{\imath}}} = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{\boldsymbol{\jmath}}} = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\, \mathbf{\hat{\boldsymbol{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 which, in the context of linear algebra, is typically referred to as a standard basis.

They are often denoted using normal vector notation (e.g., i or \vec{\imath}) rather than standard unit vector notation (e.g., \hat{\bold{\imath}}). In most contexts it can be assumed that i, j, and k, (or \vec{\imath}, \vec{\jmath}, and \vec{k}) are versors of a 3-D Cartesian coordinate system. The notations (\bold{\hat{x}}, \bold{\hat{y}}, \bold{\hat{z}}), (\bold{\hat{x}}_1, \bold{\hat{x}}_2, \bold{\hat{x}}_3), (\bold{\hat{e}}_x, \bold{\hat{e}}_y, \bold{\hat{e}}_z), or (\bold{\hat{e}}_1, \bold{\hat{e}}_2, \bold{\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).

Cylindrical coordinates

The three orthogonal unit vectors appropriate to cylindrical symmetry are:

  • \bold{\hat{s}} (also designated \bold{\hat{r}} or \boldsymbol{\hat \rho}), representing the direction along which the distance of the point from the axis of symmetry is measured;
  • \boldsymbol{\hat \varphi}, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the symmetry axis;
  • \bold{\hat{z}}, representing the direction of the symmetry axis;

They are related to the Cartesian basis \hat{x}, \hat{y}, \hat{z} by:

\bold{\hat{s}} = \cos \varphi\bold{\hat{x}} + \sin \varphi\bold{\hat{y}}
\boldsymbol{\hat \varphi} = -\sin \varphi\bold{\hat{x}} + \cos \varphi\bold{\hat{y}}
\bold{\hat{z}}=\bold{\hat{z}}.

It is important to note that \bold{\hat{s}} and \boldsymbol{\hat \varphi} are functions of \varphi, 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 matrix. The derivatives with respect to \varphi are:

\frac{\partial \bold{\hat{s}}} {\partial \varphi} = -\sin \varphi\bold{\hat{x}} + \cos \varphi\bold{\hat{y}} = \boldsymbol{\hat \varphi}
\frac{\partial \boldsymbol{\hat \varphi}} {\partial \varphi} = -\cos \varphi\bold{\hat{x}} - \sin \varphi\bold{\hat{y}} = -\bold{\hat{s}}
\frac{\partial \bold{\hat{z}}} {\partial \varphi} = \mathbf{0}.

Spherical coordinates

The unit vectors appropriate to spherical symmetry are: \bold{\hat{r}}, the direction in which the radial distance from the origin increases; \boldsymbol{\hat{\varphi}}, the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and \boldsymbol{\hat \theta}, the direction in which the angle from the positive z axis is increasing. To minimize degeneracy, the polar angle is usually taken 0\leq\theta\leq 180^\circ. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of \boldsymbol{\hat \varphi} and \boldsymbol{\hat \theta} are often reversed. Here, the American "physics" convention[1] is used. This leaves the azimuthal angle \varphi defined the same as in cylindrical coordinates. The Cartesian relations are:

\bold{\hat{r}} = \sin \theta \cos \varphi\bold{\hat{x}} + \sin \theta \sin \varphi\bold{\hat{y}} + \cos \theta\bold{\hat{z}}
\boldsymbol{\hat \theta} = \cos \theta \cos \varphi\bold{\hat{x}} + \cos \theta \sin \varphi\bold{\hat{y}} - \sin \theta\bold{\hat{z}}
\boldsymbol{\hat \varphi} = - \sin \varphi\bold{\hat{x}} + \cos \varphi\bold{\hat{y}}

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

\frac{\partial \bold{\hat{r}}} {\partial \varphi} = -\sin \theta \sin \varphi\bold{\hat{x}} + \sin \theta \cos \varphi\bold{\hat{y}} = \sin \theta\boldsymbol{\hat \varphi}
\frac{\partial \bold{\hat{r}}} {\partial \theta} =\cos \theta \cos \varphi\bold{\hat{x}} + \cos \theta \sin \varphi\bold{\hat{y}} - \sin \theta\bold{\hat{z}}= \boldsymbol{\hat \theta}
\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \varphi} =-\cos \theta \sin \varphi\bold{\hat{x}} + \cos \theta \cos \varphi\bold{\hat{y}} = \cos \theta\boldsymbol{\hat \varphi}
\frac{\partial \boldsymbol{\hat{\theta}}} {\partial \theta} = -\sin \theta \cos \varphi\bold{\hat{x}} - \sin \theta \sin \varphi\bold{\hat{y}} - \cos \theta\bold{\hat{z}} = -\bold{\hat{r}}
\frac{\partial \boldsymbol{\hat{\varphi}}} {\partial \varphi} = -\cos \varphi\bold{\hat{x}} - \sin \varphi\bold{\hat{y}} = -\sin \theta\bold{\hat{r}} -\cos \theta\boldsymbol{\hat{\theta}}

General unit vectors

Common general themes of unit vectors occur throughout physics and geometry:[2]

Unit vector Nomenclature Diagram
Tangent vector to a curve/flux line \mathbf{\hat{t}}\,\!

A normal vector \mathbf{\hat{n}} \,\! to the plane containing and defined by the radial position vector r \mathbf{\hat{r}} \,\! and angular tangential direction of rotation \theta \boldsymbol{\hat{\theta}} \,\! is necessary so that the vector equations of angular motion hold.

Normal to a surface tangent plane/plane containing radial position component and angular tangential component \mathbf{\hat{n}}\,\!

In terms of polar coordinates; \mathbf{\hat{n}} = \mathbf{\hat{r}} \times \boldsymbol{\hat{\theta}} \,\!

Binormal vector to tangent and normal \mathbf{\hat{b}} = \mathbf{\hat{t}} \times \mathbf{\hat{n}} \,\![3]
Parallel to some axis/line \mathbf{\hat{e}}_{\parallel} \,\!

One unit vector \mathbf{\hat{e}}_{\parallel}\,\! aligned parallel to a principle direction (red line), and a perpendicular unit vector \mathbf{\hat{e}}_{\bot}\,\! is in any radial direction relative to the principle line.

Perpendicular to some axis/line in some radial direction \mathbf{\hat{e}}_{\bot} \,\!
Possible angular deviation relative to some axis/line \mathbf{\hat{e}}_{\angle} \,\!

Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principle direction.

Curvilinear coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors \bold{\hat{e}}_n equal to the degrees of freedom of the space. For ordinary 3-space, these vectors may be denoted \bold{\hat{e}_1}, \bold{\hat{e}_2}, \bold{\hat{e}_3}. It is nearly always convenient to define the system to be orthonormal and right-handed:

\bold{\hat{e}_i} \cdot \bold{\hat{e}_j} = \delta_{ij}

\bold{\hat{e}_i} \cdot (\bold{\hat{e}_j} \times \bold{\hat{e}_k}) = \varepsilon_{ijk}

where δij is the Kronecker delta (which is one for i = j and zero else) and \varepsilon_{ijk} is the Levi-Civita symbol (which is one for permutations ordered as ijk and minus one for permutations ordered as kji).

See also

References

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