unit vector

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unit vector
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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": {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.(File:2D Direction Vectors.svg|thumb|Examples of two 2D direction vectors)(File:3D Direction Vectors.tiff|thumb|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.,
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

Cartesian coordinates

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
mathbf{hat{i}} = begin{bmatrix}1end{bmatrix}, ,, mathbf{hat{j}} = begin{bmatrix}01end{bmatrix}, ,, mathbf{hat{k}} = begin{bmatrix}01end{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 vec{imath}) rather than standard unit vector notation (e.g., mathbf{hat{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 (mathbf{hat{x}}, mathbf{hat{y}}, mathbf{hat{z}}), (mathbf{hat{x}}_1, mathbf{hat{x}}_2, mathbf{hat{x}}_3), (mathbf{hat{e}}_x, mathbf{hat{e}}_y, mathbf{hat{e}}_z), or (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).

Cylindrical coordinates

The three orthogonal unit vectors appropriate to cylindrical symmetry are:
  • mathbf{hat{rho}} (also designated mathbf{hat{e}} or boldsymbol{hat s}), 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;
  • mathbf{hat{z}}, representing the direction of the symmetry axis;
They are related to the Cartesian basis hat{x}, hat{y}, hat{z} by:
mathbf{hat{rho}} = cos varphimathbf{hat{x}} + sin varphimathbf{hat{y}}
boldsymbol{hat varphi} = -sin varphimathbf{hat{x}} + cos varphimathbf{hat{y}}
It is important to note that mathbf{hat{rho}} 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 mathbf{hat{rho}}} {partial varphi} = -sin varphimathbf{hat{x}} + cos varphimathbf{hat{y}} = boldsymbol{hat varphi}
frac{partial boldsymbol{hat varphi}} {partial varphi} = -cos varphimathbf{hat{x}} - sin varphimathbf{hat{y}} = -mathbf{hat{rho}}
frac{partial mathbf{hat{z}}} {partial varphi} = mathbf{0}.

Spherical coordinates

The unit vectors appropriate to spherical symmetry are: mathbf{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 redundancy of representations, the polar angle theta is usually taken to lie between zero and 180 degrees. 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" conventionTevian Dray and Corinne A. Manogue,Spherical Coordinates, College Math Journal 34, 168-169 (2003). is used. This leaves the azimuthal angle varphi defined the same as in cylindrical coordinates. The Cartesian relations are:
mathbf{hat{r}} = sin theta cos varphimathbf{hat{x}} + sin theta sin varphimathbf{hat{y}} + cos thetamathbf{hat{z}}
boldsymbol{hat theta} = cos theta cos varphimathbf{hat{x}} + cos theta sin varphimathbf{hat{y}} - sin thetamathbf{hat{z}}
boldsymbol{hat varphi} = - sin varphimathbf{hat{x}} + cos varphimathbf{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 matrix and determinant. The non-zero derivatives are:
frac{partial mathbf{hat{r}}} {partial varphi} = -sin theta sin varphimathbf{hat{x}} + sin theta cos varphimathbf{hat{y}} = sin thetaboldsymbol{hat varphi}
frac{partial mathbf{hat{r}}} {partial theta} =cos theta cos varphimathbf{hat{x}} + cos theta sin varphimathbf{hat{y}} - sin thetamathbf{hat{z}}= boldsymbol{hat theta}
frac{partial boldsymbol{hat{theta}}} {partial varphi} =-cos theta sin varphimathbf{hat{x}} + cos theta cos varphimathbf{hat{y}} = cos thetaboldsymbol{hat varphi}
frac{partial boldsymbol{hat{theta}}} {partial theta} = -sin theta cos varphimathbf{hat{x}} - sin theta sin varphimathbf{hat{y}} - cos thetamathbf{hat{z}} = -mathbf{hat{r}}
frac{partial boldsymbol{hat{varphi}}} {partial varphi} = -cos varphimathbf{hat{x}} - sin varphimathbf{hat{y}} = -sin thetamathbf{hat{r}} -cos thetaboldsymbol{hat{theta}}

General unit vectors

Common general themes of unit vectors occur throughout physics and geometry:BOOK, Calculus (Schaum's Outlines Series), 5th, Mc Graw Hill, F. Ayres, E. Mandelson, 2009, 978-0-07-150861-2, {| class="wikitable"! scope="col" width="200" | Unit vector! scope="col" width="150" | Nomenclature! scope="col" width="410" | Diagram
(File:Tangent normal binormal unit vectors.svg"200px") (File:Polar coord unit vectors and normal.svg"200px")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} times mathbf{hat{n}} VECTOR ANALYSIS (SCHAUM'S OUTLINES SERIES)>EDITION=2ND
mathbf{hat{b}} = mathbf{hatAUTHOR1=M. R. SPIEGEL AUTHOR3=D. SPELLMAN ISBN=978-0-07-161545-7,
(File:Perpendicular and parallel unit vectors.svg"200px")One unit vector mathbf{hat{e}}_{parallel} aligned parallel to a principal direction (red line), and a perpendicular unit vector mathbf{hat{e}}_{bot} is in any radial direction relative to the principal line.
| Perpendicular to some axis/line in some radial direction
| Possible angular deviation relative to some axis/line
mathbf{hat{e}}_{angle} (File:Angular unit vector.svg"200px")Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction.

Curvilinear coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors mathbf{hat{e}}_n equal to the degrees of freedom of the space. For ordinary 3-space, these vectors may be denoted mathbf{hat{e}}_1, mathbf{hat{e}}_2, mathbf{hat{e}}_3. It is nearly always convenient to define the system to be orthonormal and right-handed:
mathbf{hat{e}}_i cdot mathbf{hat{e}}_j = delta_{ij} mathbf{hat{e}}_i cdot (mathbf{hat{e}}_j times mathbf{hat{e}}_k) = varepsilon_{ijk}
where delta_{ij} is the Kronecker delta (which is 1 for i = j and 0 otherwise) and varepsilon_{ijk} is the Levi-Civita symbol (which is 1 for permutations ordered as ijk and −1 for permutations ordered as kji).

Right versor

A unit vector in ℝ3 was called a right versor by W. R. Hamilton as he developed his quaternions ℍ ⊂ ℝ4. In fact, he was the originator of the term vector as every quaternion q = s + v has a scalar part s and a vector part v. If v is a unit vector in ℝ3, then the square of v in quaternions is –1. By Euler's formula then, exp (theta v) = cos theta + v sin theta is a versor in the 3-sphere. When θ is a right angle, the versor is a right versor: its scalar part is zero and its vector part v is a unit vector in ℝ3.

See also

{{wiktionary|unit vector}}




  • BOOK, G. B. Arfken, H. J. Weber, yes, Mathematical Methods for Physicists, 5th, 2000, Academic Press, 0-12-059825-6,
  • BOOK, Murray R., Spiegel, Schaum's Outlines: Mathematical Handbook of Formulas and Tables, 2nd, 1998, McGraw-Hill, 0-07-038203-4,
  • BOOK, David J., Griffiths, Introduction to Electrodynamics, 3rd, 1998, Prentice Hall, 0-13-805326-X, registration,weblink

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