Unit vector explained

Unit vector should not be confused with Vector of ones.

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", as in

\hat{ v } (pronounced "v-hat").

The term direction vector, commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere.

The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,

$\backslash mathbf\; =\; \backslash frac$where ‖u‖ is the norm (or length) of u.^{[1] [2]} 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, and every vector in the space may be written as a linear combination form of unit vectors.

Orthogonal coordinates

Cartesian coordinates

See main article: 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 three dimensional Cartesian coordinate system are

$\backslash mathbf\; =\; \backslash begin1\backslash \backslash 0\backslash \backslash 0\backslash end,\; \backslash ,\backslash ,\; \backslash mathbf\; =\; \backslash begin0\backslash \backslash 1\backslash \backslash 0\backslash end,\; \backslash ,\backslash ,\; \backslash mathbf\; =\; \backslash begin0\backslash \backslash 0\backslash \backslash 1\backslash end$

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 $\backslash vec$) rather than standard unit vector notation (e.g., x̂). In most contexts it can be assumed that x, y, and z, (or $\backslash vec,$ $\backslash vec,$ and $\backslash vec$) are versors of a 3-D Cartesian coordinate system. The notations (î, ĵ, k̂), (x̂₁, x̂₂, x̂₃), (ê_x, ê_y, ê_z), or (ê₁, ê₂, ê₃), 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, 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 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 (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis (vector).

Cylindrical coordinates

See also: Jacobian matrix. The three orthogonal unit vectors appropriate to cylindrical symmetry are:

• $\backslash boldsymbol$ (also designated $\backslash mathbf$ or $\backslash boldsymbol$), representing the direction along which the distance of the point from the axis of symmetry is measured;
• $\backslash boldsymbol$, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the symmetry axis;
• $\backslash mathbf$, representing the direction of the symmetry axis;

They are related to the Cartesian basis $\backslash hat$, $\backslash hat$, $\backslash hat$ by:

$\backslash boldsymbol\; =\; \backslash cos(\backslash varphi)\backslash mathbf\; +\; \backslash sin(\backslash varphi)\backslash mathbf$

$\backslash boldsymbol\; =\; -\backslash sin(\backslash varphi)\; \backslash mathbf\; +\; \backslash cos(\backslash varphi)\; \backslash mathbf$

$\backslash mathbf\; =\; \backslash mathbf.$

The vectors $\backslash boldsymbol$ and $\backslash boldsymbol$ are functions of $\backslash varphi,$ and are not constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. The derivatives with respect to

\varphi are:

$\backslash frac\; =\; -\backslash sin\; \backslash varphi\backslash mathbf\; +\; \backslash cos\; \backslash varphi\backslash mathbf\; =\; \backslash boldsymbol$

$\backslash frac\; =\; -\backslash cos\; \backslash varphi\backslash mathbf\; -\; \backslash sin\; \backslash varphi\backslash mathbf\; =\; -\backslash boldsymbol$

$\backslash frac\; =\; \backslash mathbf.$

Spherical coordinates

The unit vectors appropriate to spherical symmetry are: $\backslash mathbf$, the direction in which the radial distance from the origin increases; $\backslash boldsymbol$, the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and $\backslash boldsymbol$, the direction in which the angle from the positive z axis is increasing. To minimize redundancy of representations, the polar angle $\backslash theta$ 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 $\backslash boldsymbol$ and $\backslash boldsymbol$ are often reversed. Here, the American "physics" convention^[3] is used. This leaves the azimuthal angle $\backslash varphi$ defined the same as in cylindrical coordinates. The Cartesian relations are:

$\backslash mathbf\; =\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash varphi\backslash mathbf\; +\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash varphi\backslash mathbf\; +\; \backslash cos\; \backslash theta\backslash mathbf$

$\backslash boldsymbol\; =\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash varphi\backslash mathbf\; +\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash varphi\backslash mathbf\; -\; \backslash sin\; \backslash theta\backslash mathbf$

$\backslash boldsymbol\; =\; -\; \backslash sin\; \backslash varphi\backslash mathbf\; +\; \backslash cos\; \backslash varphi\backslash mathbf$

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

$\backslash frac\; =\; -\backslash sin\; \backslash theta\; \backslash sin\; \backslash varphi\backslash mathbf\; +\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash varphi\backslash mathbf\; =\; \backslash sin\; \backslash theta\backslash boldsymbol$

$\backslash frac\; =\backslash cos\; \backslash theta\; \backslash cos\; \backslash varphi\backslash mathbf\; +\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash varphi\backslash mathbf\; -\; \backslash sin\; \backslash theta\backslash mathbf=\; \backslash boldsymbol$

$\backslash frac\; =-\backslash cos\; \backslash theta\; \backslash sin\; \backslash varphi\backslash mathbf\; +\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash varphi\backslash mathbf\; =\; \backslash cos\; \backslash theta\backslash boldsymbol$

$\backslash frac\; =\; -\backslash sin\; \backslash theta\; \backslash cos\; \backslash varphi\backslash mathbf\; -\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash varphi\backslash mathbf\; -\; \backslash cos\; \backslash theta\backslash mathbf\; =\; -\backslash mathbf$

$\backslash frac\; =\; -\backslash cos\; \backslash varphi\backslash mathbf\; -\; \backslash sin\; \backslash varphi\backslash mathbf\; =\; -\backslash sin\; \backslash theta\backslash mathbf\; -\backslash cos\; \backslash theta\backslash boldsymbol$

General unit vectors

See main article: Orthogonal coordinates.

Common themes of unit vectors occur throughout physics and geometry:^[4]

Unit vector	Nomenclature	Diagram
Tangent vector to a curve/flux line	\hat{t }	A normal vector \hat{n } to the plane containing and defined by the radial position vector r \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	\hat{n } In terms of polar coordinates; \hat{n } = \mathbf \times \boldsymbol
Binormal vector to tangent and normal	\hat{b } = \mathbf \times \mathbf ^[5]
Parallel to some axis/line	\hat{e }_	One unit vector \hat{e }_ aligned parallel to a principal direction (red line), and a perpendicular unit vector \hat{e }_ is in any radial direction relative to the principal line.
Perpendicular to some axis/line in some radial direction	\hat{e }_
Possible angular deviation relative to some axis/line	\hat{e }_	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 $\backslash mathbf\_n$ (the actual number being equal to the degrees of freedom of the space). For ordinary 3-space, these vectors may be denoted $\backslash mathbf\_1,\; \backslash mathbf\_2,\; \backslash mathbf\_3$. It is nearly always convenient to define the system to be orthonormal and right-handed:

$\backslash mathbf\_i\; \backslash cdot\; \backslash mathbf\_j\; =\; \backslash delta\_$

$\backslash mathbf\_i\; \backslash cdot\; (\backslash mathbf\_j\; \backslash times\; \backslash mathbf\_k)\; =\; \backslash varepsilon\_$

where

\delta _{ij} is the Kronecker delta (which is 1 for i = j, and 0 otherwise) and $\backslash varepsilon\_$ 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

R ^{3} was called a right versor by W. R. Hamilton, as he developed his quaternions H \subset R ^{4} . 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 R ^{3} , then the square of v in quaternions is –1. Thus by Euler's formula, \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 R ^{3} .S ^{2} \subset R ^{3} \subset H rather than the pair in the complex plane.

By extension, a right quaternion is a real multiple of a right versor.

See also

• Cartesian coordinate system
• Coordinate system
• Curvilinear coordinates
• Four-velocity
• Jacobian matrix and determinant
• Normal vector
• Polar coordinate system
• Standard basis
• Unit interval
• Unit square, cube, circle, sphere, and hyperbola
• Vector notation
• Vector of ones
• Unit matrix

References

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

Notes and References

1. Web site: Weisstein. Eric W.. Unit Vector. 2020-08-19. Wolfram MathWorld . en.
2. Web site: Unit Vectors . 2020-08-19. Brilliant Math & Science Wiki . en-us.
3. Tevian Dray and Corinne A. Manogue, Spherical Coordinates, College Math Journal 34, 168-169 (2003).
4. Book: Calculus (Schaum's Outlines Series). 5th. Mc Graw Hill. F. Ayres . E. Mendelson . 2009. 978-0-07-150861-2.
5. Book: Vector Analysis (Schaum's Outlines Series). 2nd. Mc Graw Hill. M. R. Spiegel . S. Lipschutz . D. Spellman . 2009. 978-0-07-161545-7.