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Unit vector

In mathematics and physics, a unit vector is defined as a vector in a normed vector space whose magnitude, or length, is exactly one.^[1] This property allows unit vectors to represent direction independently of scale, making them essential for normalizing other vectors or expressing components in coordinate systems.^[2] To obtain a unit vector from a nonzero vector \mathbf{v}, one divides \mathbf{v} by its magnitude \|\mathbf{v}\|, yielding \hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}, which ensures \|\hat{\mathbf{v}}\| = 1.^[3] In three-dimensional Euclidean space, the standard unit vectors are the basis vectors \mathbf{i} = (1, 0, 0), \mathbf{j} = (0, 1, 0), and \mathbf{k} = (0, 0, 1), aligned with the Cartesian axes and forming an orthonormal basis.^[4] These vectors simplify vector decomposition, such as expressing any vector \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}, where x, y, z are scalar components.^[5] Unit vectors play a crucial role in applications across physics and engineering, where they describe directions of physical quantities like velocity, force, and electric fields without specifying magnitude.^[6] For instance, in vector addition and resolution, unit vectors enable the breakdown of complex motions or forces into perpendicular components, facilitating calculations in mechanics and electromagnetism.^[7] In higher mathematics, they extend to abstract spaces, supporting concepts like orthogonality and projections in linear algebra.^[8]

Definition and Properties

Definition

A unit vector, also known as a normalized vector, is defined as any vector in a normed vector space whose magnitude, or norm, is precisely equal to 1.^[9] This property ensures that the vector captures direction without incorporating any scaling by length.^[10] The concept applies generally to any real or complex vector space equipped with a norm, though it is most frequently encountered and applied within Euclidean spaces where the norm corresponds to the standard length metric. Unit vectors are commonly denoted by a circumflex (hat) symbol over the vector, such as \hat{\mathbf{u}}, or occasionally by boldface lettering with an explicit unit indicator to emphasize their normalized status.^[9] In contrast to general vectors, which combine both magnitude and direction, a unit vector isolates pure directional information, serving as a fundamental tool for orientation without magnitude influence.^[11] For instance, in two-dimensional Euclidean space, the standard unit vector along the positive x-axis is \hat{\mathbf{i}} = (1, 0).^[12]

Normalization

The normalization of a non-zero vector \mathbf{u} to obtain a unit vector \hat{\mathbf{u}} involves scaling \mathbf{u} by the reciprocal of its magnitude, ensuring the resulting vector has length 1 while preserving direction. The formula is \hat{\mathbf{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|}, where \|\mathbf{u}\| = \sqrt{\mathbf{u} \cdot \mathbf{u}} denotes the Euclidean (L2) norm of \mathbf{u}.^[1]^[13] The process proceeds in two main steps: first, compute the magnitude \|\mathbf{u}\| by calculating the square root of the dot product of \mathbf{u} with itself; second, divide each component of \mathbf{u} by this magnitude to yield \hat{\mathbf{u}}. This operation is undefined for the zero vector, as its magnitude is zero, preventing division and confirming that no unit vector exists for it.^[13]^[14] For example, consider the 2D vector \mathbf{v} = (3, 4); its magnitude is \|\mathbf{v}\| = \sqrt{3^2 + 4^2} = 5, so the unit vector is \hat{\mathbf{v}} = \left( \frac{3}{5}, \frac{4}{5} \right). In 3D, for \mathbf{w} = (1, 1, 1), the magnitude is \|\mathbf{w}\| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}, yielding \hat{\mathbf{w}} = \frac{1}{\sqrt{3}} (1, 1, 1).^[13]^[14] In numerical computations, floating-point arithmetic can introduce precision errors during magnitude calculation, particularly when vector components vary greatly in scale, leading to loss of significant digits in the sum of squares or square root operation. While the L2 norm is standard for Euclidean unit vectors, awareness of these issues is crucial in high-dimensional or iterative algorithms to avoid accumulated inaccuracies.^[15] For any given direction in Euclidean space, exactly two unit vectors exist: one pointing in the forward direction and its opposite, obtained by multiplying by -1, both lying along the same line but with reversed orientation.^[16]

Mathematical Properties

A unit vector \hat{\mathbf{u}} in a real or complex vector space is defined to have a magnitude of exactly 1, satisfying \|\hat{\mathbf{u}}\| = 1.^[17] This property ensures that unit vectors encode pure directional information without scaling factors, distinguishing them from general vectors that may have arbitrary lengths.^[2] One key geometric role of unit vectors is in preserving and projecting directions. The scalar projection of \mathbf{v} onto \hat{\mathbf{u}} is \mathbf{v} \cdot \hat{\mathbf{u}}, and the vector projection, which is the component of \mathbf{v} along the direction of \hat{\mathbf{u}}, is (\mathbf{v} \cdot \hat{\mathbf{u}}) \hat{\mathbf{u}}. This isolates the directional contribution along that unit direction.^[18] The dot product between two unit vectors \hat{\mathbf{u}} and \hat{\mathbf{v}} simplifies to \hat{\mathbf{u}} \cdot \hat{\mathbf{v}} = \cos \theta, where \theta is the angle between them, providing a direct measure of their angular separation.^[19] Orthogonality occurs precisely when this dot product equals zero, indicating perpendicular directions (\theta = 90^\circ).^[19] A collection of pairwise orthogonal unit vectors forms an orthonormal basis for the subspace they span. In finite-dimensional spaces, such a basis \{\hat{\mathbf{e}}_1, \dots, \hat{\mathbf{e}}_n\} satisfies the completeness relation, allowing any vector \mathbf{v} in the space to be uniquely expressed as a linear combination \mathbf{v} = \sum_{i=1}^n (\mathbf{v} \cdot \hat{\mathbf{e}}_i) \hat{\mathbf{e}}_i.^[20] This decomposition leverages the orthonormality to compute coefficients directly via dot products, simplifying vector analysis and transformations.^[21] More generally, any nonzero vector \mathbf{v} can be factored as \mathbf{v} = \|\mathbf{v}\| \hat{\mathbf{v}}, where \hat{\mathbf{v}} is the corresponding unit vector in the same direction. In an orthonormal basis, this extends to full vector reconstruction, underscoring the role of unit vectors in basis expansions.^[21] For complex vector spaces, unit vectors are defined using the Hermitian inner product, satisfying \hat{\mathbf{u}}^\dagger \hat{\mathbf{u}} = 1, where \dagger denotes the conjugate transpose. This ensures the norm remains real and positive, extending the real-case properties to handle phase information in quantum mechanics and signal processing.^[22] The Hermitian dot product \hat{\mathbf{u}}^\dagger \hat{\mathbf{v}} analogously captures directional alignment, with orthogonality when it equals zero.^[23]

Unit Vectors in Orthogonal Coordinates

Cartesian Coordinates

In three-dimensional Cartesian coordinates, the standard basis unit vectors are defined as \hat{\mathbf{i}} = (1, 0, 0), \hat{\mathbf{j}} = (0, 1, 0), and \hat{\mathbf{k}} = (0, 0, 1), each pointing along the positive x-, y-, and z-axes, respectively.^[24] These vectors serve as the fundamental directions in the rectangular coordinate system. In n-dimensional Euclidean space, the standard basis extends analogously to a set of n unit vectors, where the k-th vector has a 1 in the k-th component and 0 elsewhere.^[1] Unlike unit vectors in curvilinear systems, those in Cartesian coordinates maintain constant direction and magnitude throughout the space, independent of position.^[25] They are also mutually orthogonal, with their dot products satisfying \hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = 0, \hat{\mathbf{i}} \cdot \hat{\mathbf{k}} = 0, and \hat{\mathbf{j}} \cdot \hat{\mathbf{k}} = 0, forming an orthonormal basis.^[1] A general unit vector \hat{\mathbf{u}} in three-dimensional Cartesian coordinates can be expressed as a linear combination \hat{\mathbf{u}} = u_x \hat{\mathbf{i}} + u_y \hat{\mathbf{j}} + u_z \hat{\mathbf{k}}, where the components satisfy the normalization condition u_x^2 + u_y^2 + u_z^2 = 1.^[26] This representation ensures \hat{\mathbf{u}} has magnitude 1 while pointing in an arbitrary direction within the space. These unit vectors enable the decomposition of any vector \mathbf{v} into \mathbf{v} = v_x \hat{\mathbf{i}} + v_y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}}, where the components v_x, v_y, v_z are obtained via simple projections such as \mathbf{v} \cdot \hat{\mathbf{i}} = v_x.^[24] This facilitates straightforward calculations in vector algebra and calculus, such as resolving forces or velocities along coordinate axes. The Cartesian coordinate system, including its unit vectors, originated in René Descartes' 1637 work La Géométrie, which linked algebraic equations to geometric points via rectangular axes.^[27] The use of unit vectors as basis elements was formalized within vector calculus by J. Willard Gibbs and Oliver Heaviside in the late 19th century, providing a rigorous framework for physical applications.^[28]

Cylindrical Coordinates

In cylindrical coordinates, a point in three-dimensional space is specified by the radial distance \rho from the z-axis, the azimuthal angle \phi measured from the positive x-axis in the xy-plane, and the height z along the z-axis, denoted as (\rho, \phi, z).^[29] This system extends polar coordinates into three dimensions by adding the z-component, useful for problems with cylindrical symmetry.^[30] The orthonormal basis consists of three unit vectors: \hat{\boldsymbol{\rho}}, \hat{\boldsymbol{\phi}}, and \hat{\mathbf{z}}. Expressed in Cartesian coordinates, these are \hat{\boldsymbol{\rho}} = (\cos \phi, \sin \phi, 0), \hat{\boldsymbol{\phi}} = (-\sin \phi, \cos \phi, 0), and \hat{\mathbf{z}} = (0, 0, 1).^[31] The vectors \hat{\boldsymbol{\rho}} and \hat{\boldsymbol{\phi}} depend on the angle \phi, pointing radially outward and in the direction of increasing \phi, respectively, while \hat{\mathbf{z}} remains constant along the z-direction.^[32] These unit vectors are mutually orthogonal at every point, satisfying \hat{\boldsymbol{\rho}} \cdot \hat{\boldsymbol{\phi}} = 0, \hat{\boldsymbol{\rho}} \cdot \hat{\mathbf{z}} = 0, and \hat{\boldsymbol{\phi}} \cdot \hat{\mathbf{z}} = 0, forming a right-handed basis.^[30] The position vector of a point in cylindrical coordinates is given by \mathbf{r} = \rho \hat{\boldsymbol{\rho}} + z \hat{\mathbf{z}}, omitting the \phi-component since it does not contribute to displacement along the angular direction.^[25] This representation arises from a transformation of the Cartesian position vector \mathbf{r} = x \hat{\mathbf{i}} + y \hat{\mathbf{j}} + z \hat{\mathbf{k}}, where the cylindrical basis is obtained via a rotation matrix that depends on \phi:

\begin{pmatrix} \hat{\boldsymbol{\rho}} \\ \hat{\boldsymbol{\phi}} \\ \hat{\mathbf{z}} \end{pmatrix} = \begin{pmatrix} \cos \phi & \sin \phi & 0 \\ -\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \hat{\mathbf{i}} \\ \hat{\mathbf{j}} \\ \hat{\mathbf{k}} \end{pmatrix}.

^[31] This angular dependence distinguishes cylindrical unit vectors from the fixed Cartesian basis, enabling efficient description of rotationally symmetric fields.^[33]

Spherical Coordinates

In spherical coordinates, a point in three-dimensional Euclidean space is identified by the triplet (r, \theta, \phi), where r \geq 0 is the radial distance from the origin, \theta \in [0, \pi] is the polar angle measured from the positive z-axis, and \phi \in [0, 2\pi) is the azimuthal angle in the xy-plane from the positive x-axis.^[34] This system is particularly suited for problems exhibiting spherical symmetry, such as gravitational or electrostatic fields around a point source.^[35] The orthonormal basis of unit vectors in this system, denoted \hat{\mathbf{r}}, \hat{\boldsymbol{\theta}}, and \hat{\boldsymbol{\phi}}, point in the directions of increasing r, \theta, and \phi, respectively. Expressed in Cartesian coordinates, these are:

\hat{\mathbf{r}} = \sin\theta \cos\phi \, \hat{\mathbf{x}} + \sin\theta \sin\phi \, \hat{\mathbf{y}} + \cos\theta \, \hat{\mathbf{z}},

\hat{\boldsymbol{\theta}} = \cos\theta \cos\phi \, \hat{\mathbf{x}} + \cos\theta \sin\phi \, \hat{\mathbf{y}} - \sin\theta \, \hat{\mathbf{z}},

\hat{\boldsymbol{\phi}} = -\sin\phi \, \hat{\mathbf{x}} + \cos\phi \, \hat{\mathbf{y}}.

^[34] Unlike the constant unit vectors in Cartesian coordinates, these spherical unit vectors depend on both the polar angle \theta and the azimuthal angle \phi, reflecting the curvature of the coordinate surfaces.^[34] This angular dependence arises from the geometry of the sphere, where the directions of increasing \theta and \phi rotate as the position changes. For instance, \hat{\mathbf{r}} aligns radially outward but its orientation varies continuously with \theta and \phi. In visualization, all three unit vectors shift with position to maintain orthogonality on the local tangent plane, contrasting with cylindrical coordinates where the axial unit vector remains fixed.^[34] Any vector field in spherical coordinates can be decomposed using these unit vectors, such as \mathbf{A} = A_r \hat{\mathbf{r}} + A_\theta \hat{\boldsymbol{\theta}} + A_\phi \hat{\boldsymbol{\phi}}, where the components A_r, A_\theta, and A_\phi are scalar functions of position. This decomposition is essential for fields with radial symmetry, like the electric field \mathbf{E} around a point charge, expressed as \mathbf{E} = E_r \hat{\mathbf{r}} + E_\theta \hat{\boldsymbol{\theta}} + E_\phi \hat{\boldsymbol{\phi}}.^[35] The transformation between spherical and Cartesian coordinates involves a Jacobian matrix whose determinant is r^2 \sin \theta, incorporating the angular dependencies from spherical geometry; this factor scales differentials in integrals, such as the volume element dV = r^2 \sin \theta \, dr \, d\theta \, d\phi.^[36]

General Form

In orthogonal curvilinear coordinate systems, the unit vectors \hat{\mathbf{e}}_i are defined as the normalized tangent vectors to the coordinate curves, where the position vector \mathbf{r} is expressed as a function of the coordinates q_1, q_2, \dots, q_n. The scale factor h_i for each coordinate direction is given by h_i = \left| \frac{\partial \mathbf{r}}{\partial q_i} \right|, and the unit vector is \hat{\mathbf{e}}_i = \frac{1}{h_i} \frac{\partial \mathbf{r}}{\partial q_i}.^[37] These unit vectors form an orthonormal basis, satisfying \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j = \delta_{ij}, where \delta_{ij} is the Kronecker delta (equal to 1 if i = j and 0 otherwise). This orthogonality arises because the coordinate system is designed such that the tangent vectors to different coordinate curves are perpendicular at each point.^[38] The geometry of the space is described by the line element ds^2 = \sum_i h_i^2 dq_i^2, which corresponds to a diagonal metric tensor g_{ij} = h_i^2 \delta_{ij} in the orthogonal basis. This form encapsulates the infinitesimal distances along each coordinate direction, scaled appropriately by the local geometry.^[39] This general framework extends naturally to arbitrary dimensions n > 3, where the orthonormal set \{\hat{\mathbf{e}}_i\}_{i=1}^n provides a local basis for vector decomposition in the coordinate space. It unifies specific orthogonal systems, such as Cartesian coordinates where all h_i = 1, cylindrical coordinates with h_\rho = 1, h_\phi = \rho, h_z = 1, and spherical coordinates with h_r = 1, h_\theta = r, h_\phi = r \sin \theta, by abstracting the normalization via scale factors.^[40]

Unit Vectors in Curvilinear Coordinates

Local Tangent Basis

In general curvilinear coordinate systems, denoted by parameters (q^1, \dots, q^n), the position vector in Euclidean space is expressed as \mathbf{r} = \mathbf{r}(q^1, \dots, q^n). This parametrization defines a mapping from the coordinate domain to the ambient space, where curves of constant coordinates except one trace out the coordinate lines.^[41] The natural tangent vectors to these coordinate lines are given by \mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial q^i} for i = 1, \dots, n, which point along the directions of increasing q^i while holding other coordinates fixed. These \mathbf{e}_i form a local covariant basis at each point, but they are generally neither unit length nor orthogonal. To obtain unit vectors, normalize them as \hat{\mathbf{e}}_i = \frac{\mathbf{e}_i}{\|\mathbf{e}_i\|}, where \|\mathbf{e}_i\| = \sqrt{\mathbf{e}_i \cdot \mathbf{e}_i}. The resulting \{\hat{\mathbf{e}}_i\} constitute the local tangent basis of unit vectors, which varies continuously with position due to the curvature of the coordinate lines.^[37]^[42] In general, this local tangent basis is oblique, meaning \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j \neq 0 for i \neq j unless the coordinate system is orthogonal. The geometry is captured by the metric tensor g_{ij} = \mathbf{e}_i \cdot \mathbf{e}_j, which determines the infinitesimal line element ds^2 = g_{ij} \, dq^i \, dq^j (using Einstein summation convention). This metric encodes both the lengths \|\mathbf{e}_i\| along coordinate directions and the angles between them, highlighting the non-constancy of the unit basis across the space.^[43]^[41] An illustrative example occurs in toroidal coordinates, where the unit tangent vectors \hat{\mathbf{e}}_i twist and rotate as one moves along the toroidal surfaces, adapting to the ring-like geometry and ensuring the basis remains tangent to the local coordinate curves at every point.

Variation and Scale Factors

In curvilinear coordinates, the unit basis vectors \hat{\mathbf{e}}_i depend on position, meaning their partial derivatives with respect to the coordinates q^j are generally nonzero: \frac{\partial \hat{\mathbf{e}}_i}{\partial q^j} \neq 0.^[44] This position dependence complicates vector analysis and leads to the introduction of Christoffel symbols in differential geometry, which quantify how the basis vectors change along coordinate directions.^[44] Building on the local tangent basis defined by partial derivatives of the position vector, these variations must be accounted for in operations like differentiation to maintain coordinate invariance.^[45] Scale factors play a crucial role in normalizing the basis vectors to unit length. Defined as h_i = \|\mathbf{e}_i\|, where \mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial q^i} is the tangent vector to the q^i-coordinate curve, the unit vectors are obtained via \hat{\mathbf{e}}_i = \mathbf{e}_i / h_i.^[45] These scale factors themselves vary with position, reflecting the local stretching or compression of the coordinate grid. For explicit computation, consider elliptic cylindrical coordinates (u, v, z), where the position vector is \mathbf{r} = a \cosh u \cos v \, \hat{\mathbf{x}} + a \sinh u \sin v \, \hat{\mathbf{y}} + z \, \hat{\mathbf{z}} and a is the interfocal distance; the scale factors are h_u = h_v = a \sqrt{\cosh^2 u - \cos^2 v} and h_z = 1.^[46] Thus, the unit vectors \hat{\mathbf{e}}_u and \hat{\mathbf{e}}_v are normalized by dividing the respective tangent vectors by this common scale factor, which depends on both u and v. This normalization extends to differential operators, such as the gradient, which incorporates scale factors to express directional derivatives correctly: \nabla f = \sum_i \frac{1}{h_i} \frac{\partial f}{\partial q^i} \hat{\mathbf{e}}_i.^[45] Similarly, the divergence and curl formulas adjust for these factors and the unit vector variations to preserve physical meaning. In electromagnetism, such position-dependent unit vectors influence field expressions; for instance, when computing the curl of the magnetic field in cylindrical coordinates, the derivatives of the azimuthal unit vector \hat{\boldsymbol{\phi}} contribute additional terms that affect flux calculations through curved surfaces.^[47] A concrete illustration of unit vector variation occurs in cylindrical coordinates (\rho, \phi, z), where \hat{\boldsymbol{\rho}} = \cos \phi \, \hat{\mathbf{x}} + \sin \phi \, \hat{\mathbf{y}}. Differentiating with respect to the azimuthal angle yields \frac{\partial \hat{\boldsymbol{\rho}}}{\partial \phi} = -\sin \phi \, \hat{\mathbf{x}} + \cos \phi \, \hat{\mathbf{y}} = \hat{\boldsymbol{\phi}}, demonstrating how rotation along \phi rotates the radial unit vector into the tangential direction.^[48] This nonzero derivative highlights the need for careful handling in vector calculus applications, such as deriving Maxwell's equations in non-Cartesian systems.^[47]

Versors

A versor is a product of invertible vectors within a Clifford algebra or division algebra, such as the quaternion algebra.^[49] Unit versors have norm 1. In the specific case of quaternions, versors are unit quaternions that represent rotations in three-dimensional space.^[50] This multiplicative structure extends the concept of unit vectors beyond additive Euclidean geometry into non-commutative algebraic operations.^[51] The term "versor" was coined by William Rowan Hamilton in the 1840s during his development of quaternion theory, originally defining it as the quotient of two directed lines or vectors of equal length. Hamilton introduced quaternions on October 16, 1843, to handle three-dimensional rotations, and versors emerged as key elements for describing oriented turns.^[52] Within this framework, a "right versor" specifically denotes a versor that effects a right-angle rotation while preserving right-handed orientation in vector products, aligning with the handedness of the quaternion basis elements i, j, k. In quaternion representation, a versor is a unit quaternion q satisfying |q| = 1, where the norm is |q| = \sqrt{q \bar{q}}.^[53] Such a versor induces a rotation on a pure vector \mathbf{v} (a quaternion with zero scalar part) via the conjugation formula q \mathbf{v} q^{-1}, which preserves the vector's magnitude and represents a rotation by twice the argument of q around its vector part axis.^[54] This operation leverages the non-commutative multiplication of quaternions to model spatial transformations efficiently. Pure imaginary unit quaternions, of the form $0 + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} where x^2 + y^2 + z^2 = 1, directly correspond to spatial unit vectors, bridging the geometric interpretation of unit vectors with algebraic versors. A representative example is the versor for rotation around a unit axis \hat{\mathbf{n}} by angle \theta, given by

q = \cos\left(\frac{\theta}{2}\right) + \sin\left(\frac{\theta}{2}\right) \hat{\mathbf{n}},

where \hat{\mathbf{n}} is expressed as a pure imaginary quaternion.^[55] This form ensures |q| = 1 and facilitates smooth interpolation in applications like computer graphics, where versors enable gimbal-lock-free rotations and are widely used in animation and simulation software.^[54]

Normalization in Vector Spaces

In a normed vector space V equipped with a norm \|\cdot\|, a unit vector is defined as any vector \mathbf{u} \in V such that \|\mathbf{u}\| = 1. This generalizes the concept beyond Euclidean spaces, allowing unit vectors with respect to various norms, such as the p-norms on \mathbb{R}^n given by \|\mathbf{x}\|_p = \left( \sum_{i=1}^n |x_i|^p \right)^{1/p} for $1 \leq p < \infty, or the infinity norm \|\mathbf{x}\|_\infty = \max_i |x_i|. To obtain a unit vector from a nonzero vector \mathbf{v} \in V, one normalizes by computing \hat{\mathbf{v}} = \mathbf{v} / \|\mathbf{v}\|, ensuring \|\hat{\mathbf{v}}\| = 1. For instance, in the \ell^1 norm, normalization scales the vector so the sum of absolute components equals 1, which is useful in probability distributions or sparse signal processing.^[10]^[56]^[56] In inner product spaces, which are normed via the induced norm \|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}, unit vectors play a key role in constructing orthonormal bases. The Gram-Schmidt process transforms a linearly independent set \{\mathbf{v}_1, \dots, \mathbf{v}_k\} into an orthonormal set \{\mathbf{u}_1, \dots, \mathbf{u}_k\} by iteratively orthogonalizing and normalizing: starting with \mathbf{u}_1 = \mathbf{v}_1 / \|\mathbf{v}_1\|, then \mathbf{u}_j = (\mathbf{v}_j - \sum_{i=1}^{j-1} \langle \mathbf{v}_j, \mathbf{u}_i \rangle \mathbf{u}_i) / \|\cdot\| for j > 1. This yields vectors satisfying \langle \mathbf{u}_i, \mathbf{u}_j \rangle = \delta_{ij}, forming a basis where each \mathbf{u}_i is a unit vector. In finite-dimensional spaces, every such space admits an orthonormal basis via this method.^[57]^[58]^[57] Hilbert spaces extend this to infinite dimensions, where complete inner product spaces like L^2([a,b]) feature unit vectors in function spaces. Normalization here produces \hat{f}(x) = f(x) / \|f\|_{L^2}, with \|f\|_{L^2} = \sqrt{\int_a^b |f(x)|^2 \, dx}, ensuring the normalized function has unit L^2-norm. Orthonormal bases exist in separable Hilbert spaces, often via Gram-Schmidt on countable dense sets, as in the Fourier basis for L^2([-\pi, \pi]). Applications include the spectral theorem for self-adjoint operators on Hilbert spaces, which decomposes the space into an orthonormal basis of unit eigenvectors, facilitating diagonalization and quantum mechanical state representations.^[59] In machine learning, unit vector normalization of feature vectors—often via \ell^2-norm scaling to \hat{\mathbf{x}} = \mathbf{x} / \|\mathbf{x}\|_2—standardizes inputs for algorithms sensitive to scale, such as support vector machines or neural networks, improving convergence and performance. For the sup norm, normalization sets the maximum absolute component to 1, as in \hat{\mathbf{u}} = \mathbf{u} / \|\mathbf{u}\|_\infty, which bounds features in [−1,1] and aids robustness in high-dimensional data. In higher dimensions, the set of unit vectors in \mathbb{R}^n forms the unit sphere S^{n-1} = \{\mathbf{u} \in \mathbb{R}^n : \|\mathbf{u}\| = 1\}, a hypersurface whose geometry influences concentration phenomena and sampling in statistics.^[60]^[61]^[56]^[62]

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Dec 3, 2008 · When the right-angled coordinate system of algebraic geometry was subsequently developed, the name “Cartesian coordinates” honored Descartes' ...
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[PDF] A Historical Study of Vector Analysis - Deep Blue Repositories
In the list of notations, we notice that Heaviside used the quaternion notation for the curl even though he was opposed to quaternion analysis. In one of his ...
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[PDF] notes.coordinates.pdf - OSU Math
Cylindrical coordinates (r, θ, z) are related to Cartesian coordinates by r = px² + y², θ = arctan(y/x), z = z. Unit vectors er and eθ are defined.<|control11|><|separator|>
[30]
Cylindrical Coordinates - Richard Fitzpatrick
Of course, the unit basis vectors $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_z$ are mutually orthogonal, so $ A_r = {\bf A}\cdot {\bf e}_r , et cetera ...
[31]
[PDF] [gex67] Unit vectors of cylindrical coordinates
(a) Express the unit vectors e1 = ˆρ, e2 = ˆφ, e3 = z of cylindrical coordinates as linear combinations of the unit vectors î, j, k of Cartesian coordinates.
[32]
[PDF] Cylindrical Coordinates
The unit vectors in the cylindrical coordinate system are functions of position. It is convenient to express them in terms of the cylindrical coordinates and ...
[33]
Cylindrical coordinates - Dynamics
A point P at a time-varying position (r,θ,z) ( r , θ , z ) has position vector ⃗ρ , velocity ⃗v=˙⃗ρ v → = ρ → ˙ , and acceleration ⃗a=¨⃗ρ a → = ρ → ¨ given by the ...
[34]
1 Appendix: Vector Operations - MIT
In this text, use is made of the Cartesian, circular cylindrical, and spherical coordinate systems. In these three-dimensional systems, any vector is completely ...
[35]
Spherical Coordinates - Richard Fitzpatrick
Of course, the unit vectors $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_\phi$ are mutually orthogonal, so $ A_r = {\bf A}\cdot {\bf e}_r , et cetera ...
[36]
The Jacobian for Polar and Spherical Coordinates
The Jacobian is computed for the change from Cartesian to polar coordinates, and then for the change from Cartesian to spherical coordinates.
[37]
[PDF] MATH 2443–008 Calculus IV Spring 2014
Since these coordinate curves are not usually straight lines, the coordinates are called curvilinear. and define the unit vectors ûi by ûi = 1 hi ∂r ∂ui We ...
[38]
[PDF] AN INTRODUCTION TO CURVILINEAR ORTHOGONAL ...
This is the orthogonality property of vectors, and orthogonal coordinate systems are those in which all the unit vectors obey this property. We will learn ...
[39]
Orthogonal Curvilinear Coordinates - Richard Fitzpatrick
It follows that $ u_1$ , $ u_2$ , $ u_3$ can be used as an alternative set of coordinates to distinguish different points in space.
[40]
[PDF] A.7 ORTHOGONAL CURVILINEAR COORDINATES
" In general, these vectors are not orthogonal to one another and they are not of unit length. However, their cross product is orthogonal to by. De an clo z ...
[41]
[PDF] FW Math 321, 10/01/2003 Curvilinear Coordinates Let x, y and z be ...
Fixing q2,q3 and varying q1, the vector function r = r(q1,q2,q3) describes a curve. The derivative. ∂r/∂q1 is tangent to that curve.<|control11|><|separator|>
[42]
[PDF] Physics 504, Lecture 4 Feb. 1, 2010 1 Curvilinear Coordinates
Feb 1, 2010 · It is in general a nontrivial function of the position, gij(q). To repeat: the metric tensor determines the distance between neighboring points,.
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[PDF] An Introduction to Vectors and Tensors from a Computational ... - UTC
This introduces a dual system of local base vectors consisting of a) the tangent (covariant) base vectors that are tangent to the coordinate directions, and b) ...Missing: geometry | Show results with:geometry
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[PDF] Curvilinear Analysis in a Euclidean Space
Jun 17, 2004 · Curvilinear analysis in a Euclidean space is presented for students and professionals familiar with Cartesian analysis in 3D physical ...
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[PDF] COORDINATE SYSTEMS , ox ox - Math@LSU
A cartesian coordinate system offers the unique advantage that all three unit vectors, i, j, and k, are constant. We did introduce the radial distance r but.
[46]
Elliptic Cylindrical Coordinates -- from Wolfram MathWorld
z. (20). Then the new scale factors are. h_(q_1), = asqrt((q_1^2-q_2^2)/(q_1^2 ... References. Arfken, G. "Elliptic Cylindrical Coordinates ( u , v , z ) ...
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[PDF] PH 2301 Electromagnetic Fields I: - Curvilinear coordinates - WPI
Because the unit vectors in curvilinear coordinates are position-dependent you can not blindly take these unit vectors outside an integral or a derivative ...
[48]
[PDF] Unit 9 Vector Fields and Curvilinear Coordinates
The unit vectors ˆr , ˆs , ˆ! , and ˆ! are position-dependent. Cartesian unit vectors always point in the same direction, but curvilinear unit vectors do not.Missing: scale | Show results with:scale
[49]
[PDF] Geometric Algebra - arXiv
May 27, 2012 · Typically a rotor is the product of unit vectors, in which case ... This also means that the product of a non-null versor and any versor is ...
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[PDF] Let's have fun with Geometric algebra Jirka Velebil Department of ...
Nov 7, 2014 · Recall the reflection through a plane. In general, a versor is a product of unit vectors. 2. A product of two unit vectors is ...
[51]
[PDF] Clifford Algebra to Geometric Calculus - MIT Mathematics
... product of unit vectors; thus,. 82k/2 e. =Ulk-lUlk'. (5.29). This factorization ... versor which can be factored into a product of k vectors will be called ...
[52]
[PDF] Summary1 of Quaternions. - USC Dornsife
N(q). (if N(q) > 0). A versor is a quaternion q with N(q) = 1; it is used to describe rotations in R3. A major algebraic miracle (which, of course, follows ...
[53]
[PDF] MATH431: Quaternions - UMD MATH
Dec 5, 2021 · This is very important because when discussing rotations we can say that an arbitrary rotation can be performed via v 7→ pvp∗ where p is a unit ...
[54]
[PDF] Quaternions and Rotations∗
Sep 10, 2013 · The quaternion operator Lq* (v) = q∗vq may be interpreted as a coordinate frame rotation with respect to the (fixed) space of points. Let p and ...
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[PDF] Universal approach to derivation of quaternion rotation formulas
The rotation axis represents the unit quaternion n = 1 i + 0 j + 0 k while the rotation operator is given by q = cos α 2 + n · sin α 2 = cos α 2 + i · sin α 2 ...
[56]
[PDF] 1 Norms and Vector Spaces
Suppose we have a complex vector space V . A norm is a function f : V → R which satisfies. (i) f(x) ≥ 0 for all x ∈ V. (ii) f(x + y) ≤ f(x) + f(y) for all x ...
[57]
Gram-Schmidt process - StatLect
Every inner product space has an orthonormal basis. The following proposition presents an important consequence of the Gram-Schmidt process. Proposition Let ...Preliminaries · Projections on orthonormal sets · Every inner product space has...
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[PDF] Inner Product Space / Gram-Schmidt Process (textbook § 6.1.6.2)
Def": An orthonormal basis B for an inner product space an ordered basis which is orthonormal. Example: (1) B={e₁, ez, en} the standard basis is orthonormal ...
[59]
[PDF] The Spectral Theorem Let V be a finite-dimensional inner-product ...
If T is a normal operator then eigenvectors of T corresponding to distinct eigenvalues are orthogonal. This is true even if V is infinite-dimensional. Indeed, ...
[60]
Numerical data: Normalization | Machine Learning
Aug 25, 2025 · The goal of normalization is to transform features to be on a similar scale. For example, consider the following two features: Feature X spans ...
[61]
What is Normalization in Machine Learning? A ... - DataCamp
Jan 4, 2024 · Normalization is a specific form of feature scaling that transforms the range of features to a standard scale.
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[PDF] Volume of a ball of radius R in Rn - OU Math
Let Vn(R) and An(R) stand for the volume of Bn(R) and the area of Sn(R), respectively: Vn(R) := Volume of Bn(R) ,. An(R) := Area of Sn(R) . Observation: Just by ...