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Vector of length one
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
Unit_vector
Geometric object that has length and direction
Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement
Euclidean_vector
Use of coordinates for representing vectors
Vector notation In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more
Vector_notation
Direction and rate of rotation
because r̂ is a radial unit vector; and the perpendicular component is given by rφ˙ because φ̂ is a perpendicular unit vector. In two dimensions, angular
Angular_velocity
Computer processor which works on arrays of several numbers at once
In computing, a vector processor is a central processing unit (CPU) that implements an instruction set where its instructions are designed to operate
Vector_processor
Mathematical notation
{b} .} 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
Hat_notation
Parameterization of a rotation into a unit vector and angle
rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation
Axis–angle_representation
Property shared by codirectional lines
starting positions, defining different unit directed line segments (as a bound vector instead of a free vector). Two colinear rays or oriented line segments
Direction_(geometry)
Formulas in differential geometry
defined as follows: T is the unit vector tangent to the curve, pointing in the direction of motion. N is the normal unit vector, the derivative of T with
Frenet–Serret_formulas
Method used to normalize the range of independent variables
feature. Unit vector normalization regards each individual data point as a vector, and divide each by its vector norm, to obtain x′=x/‖x‖. Any vector norm
Feature_scaling
Vectors whose components are all 0 except one that is 1
emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely
Standard_basis
Mathematical operation on vectors in 3D space
lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly
Cross_product
Hamilton's original treatment of quaternions
and negative if they are anti-parallel. A unit vector is a vector of length one. Examples of unit vectors include i, j and k. Note: The use of the word
Classical Hamiltonian quaternions
Classical_Hamiltonian_quaternions
Multivariate derivative (mathematics)
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) ∇f whose
Gradient
Property of two or more vectors that are orthogonal and of unit length
algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of
Orthonormality
Repeating unit formed by the vectors spanning the points of a lattice
a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector
Unit_cell
Concept in linear algebra
The vector projection, also known as the vector component or vector resolution, of a vector a on (or onto) a non-zero vector b is the orthogonal projection
Vector_projection
Physical quantity that is a vector
a unit of measurement and a vector numerical value (unitless), often a Euclidean vector with magnitude and direction. For example, a position vector in
Vector_quantity
Broad concept generalizing scalars in mathematics and physics
a unit of measurement and a vector numerical value (unitless), often a Euclidean vector with magnitude and direction. For example, a position vector in
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Line or vector perpendicular to a curve or a surface
normal vector is a vector perpendicular to a given object at a particular point. A normal vector of length one is called a unit normal vector or normal
Normal_(geometry)
Assignment of a vector to each point in a subset of Euclidean space
point on the plane. Vector fields often have unit of measurement (for example, metres or kilometres per hour), forming a vector physical quantity. They
Vector_field
Amount of charge flowing through a unit cross-sectional area per unit time
amount of charge per unit time) that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude
Current_density
Instantaneous rate of change of the function
of a function per unit distance in that direction. In that convention the nonzero vector v is first normalized to the unit vector 𝐯̂=𝐯/‖𝐯‖, where
Directional_derivative
Cosines of the angles between a vector and the coordinate axes
contributions of each component of the basis to a unit vector in that direction. If v is a Euclidean vector in three-dimensional Euclidean space, R 3 ,
Direction_cosine
Central processing unit by Sony Computer Entertainment and Toshiba
separate "units", each performing a specific task, integrated onto the same die. These units are: a CPU core, two Vector Processing Units (VPU), a 10-channel
Emotion_Engine
Similarity measure for number sequences
the magnitude of the vector, yielding a vector of unit length. Then the Euclidean distance over the end-points of any two vectors is a proper metric which
Cosine_similarity
Distance from origin of tangent hyperplanes
the support function is most intuitive when x {\displaystyle x} is a unit vector. By definition, the convex set A {\displaystyle A} is contained in the
Support_function
Mathematical concept applicable to physics
in applied mathematics and vector calculus which has many applications in physics. For transport phenomena, flux is a vector quantity, describing the magnitude
Flux
Four-dimensional number system
taken to represent vectors in 3D space, then it turns out that the reflection of a vector r in a plane perpendicular to a unit vector w can be written:
Quaternion
Force directed to the center of rotation
showing that the vector dot product is zero. The unit magnitude of these vectors is a consequence of Eq. 1. Using the tangent vector, the angle θ of the
Centripetal_force
Tensor that rotates the reference frame to simplify analysis
{u}}_{D}} unit vectors (i.e., the angle between the two reference frames). The projection of the arbitrary vector onto each of the two new unit vectors implies
Direct-quadrature-zero transformation
Direct-quadrature-zero_transformation
Matrix consisting of a single row or column
vector for input to matrix transformation. Covariance and contravariance of vectors Index notation Vector of ones Single-entry vector Standard unit vector
Row_and_column_vectors
Method of data analysis
sequence of p unit vectors, where the i-th vector is the direction of a line that best fits the data while being orthogonal to the first i−1 vectors. Here, a
Principal_component_analysis
Way to determine a preliminary orbit from initial observations in astronomy
product of vectors 𝐚 and 𝐛 D0 is the common scalar triple product 𝐩𝐧 is the respective cross product vector 𝝆̂𝐧 is the respective unit vector Calculate
Gauss's_method
Position of something in relation to its surroundings
in geology and grade on maps and signs. A unit vector may also be used to represent an object's normal vector direction or the relative direction between
Orientation_(geometry)
Index of articles associated with the same name
between the two vectors. So, if n ^ {\displaystyle \mathbf {\hat {n}} } is the unit vector perpendicular to the plane determined by vectors a {\displaystyle
Vector_multiplication
Vector describing a wave; often its propagation direction
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction
Wave_vector
and their notations. Note that bold text indicates that the quantity is a vector. List of letters used in mathematics and science Glossary of mathematical
List of common physics notations
List_of_common_physics_notations
Quaternion of norm 1 (unit quaternion)
quotient of two vectors. A versor can be defined as the quotient of two unit vectors. For any fixed plane Π the quotient of two unit vectors lying in Π depends
Versor
Matrix decomposition
of orthonormal vectors, which can be regarded as basis vectors. The matrix 𝐌 maps the basis vector 𝐯i to the stretched unit vector σi𝐮i. By the
Singular_value_decomposition
Concept in 3-dimensional geometry
For a finite planar surface of scalar area S and unit normal ^n, the vector area S is defined as the unit normal scaled by the area: S = n ^ S {\displaystyle
Vector_area
Representation of mechanical stress at every point within a deformed 3D object
i j {\displaystyle \sigma _{ij}} and relates a unit-length direction vector e to the traction vector T(e) across a surface perpendicular to e: T ( e
Cauchy_stress_tensor
Physical field surrounding an electric charge
permittivity of free space. 𝐫̂01 is a unit vector directed from 𝐫1 to 𝐫0. 𝐫01 is the displacement vector from 𝐫1 to 𝐫0. For free point charges
Electric_field
Ways to represent 3D rotations
needs to track a target. Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
Vector in relativity
In special relativity, a four-vector (or 4-vector, sometimes Lorentz vector) is an element of a four-dimensional vector space object with four components
Four-vector
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
vector is directed along the unit vector k {\displaystyle \mathbf {k} } which is perpendicular to the plane of movement. Introduce the unit vectors e
Moment_of_inertia
Concept in mathematics
matrix unit with i = 1 and j = 2 is E 12 = [ 0 1 0 0 0 0 0 0 0 ] {\displaystyle E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}} A vector unit is
Matrix_unit
Vector used in astronomy
In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one
Laplace–Runge–Lenz_vector
Second order tensor in vector algebra
that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar
Dyadics
Measure of directional electromagnetic energy flux
physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area, per unit time) or power flow
Poynting_vector
Orthogonality of the directions of the principal curvatures of a surface
not all κX are equal, there is some unit vector X1 for which k1 = κX1 is as large as possible, and another unit vector X2 for which k2 = κX2 is as small
Euler's theorem (differential geometry)
Euler's_theorem_(differential_geometry)
Artistic concept relating to perspective
f/h) be the unit vector associated with q, where h = √x2 + y2 + f2. If we consider a straight line in space S with the unit vector ns ≡ (nx, ny, nz)
Vanishing_point
Vector field representation in 3D curvilinear coordinate systems
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space. When these spaces are in (typically) three dimensions
Vector fields in cylindrical and spherical coordinates
Vector_fields_in_cylindrical_and_spherical_coordinates
Measurable property of a material or system
the numerical value and kg is the unit symbol (for kilogram). Vector quantities have, besides numerical value and unit, direction or orientation in space
Physical_quantity
Vector representing the position of a point with respect to a fixed origin
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space.
Position_(geometry)
Speed and direction of a motion
(International System of Units) system. For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change
Velocity
Mathematical identities
unit vectors for the x, y, z-axes. More generally, for a function of n variables ψ(x1,…,xn), also called a scalar field, the gradient is the vector field:
Vector_calculus_identities
Generalization of the one-dimensional normal distribution to higher dimensions
normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination
Multivariate normal distribution
Multivariate_normal_distribution
Property of space that quantifies the magnetic influence at a given location
Definition of the H field (vector form, SI units) 𝐇≡1μ0𝐁−𝐌, where μ0 is the vacuum permeability, and M is the magnetization vector which represents how magnetized
Magnetic_field
Frame-dependent apparent force in Physics
\end{aligned}}} with uθ a unit vector perpendicular to uR at time t (as can be verified by noticing that the vector dot product with the radial vector is zero) and
Fictitious_force
Algebraic operation on coordinate vectors
numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their
Dot_product
Vector field that is the gradient of some function
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property
Conservative_vector_field
be placed, describing a scene for 3D rendering. 3D unit vector A unit vector in 3D space. 4D vector A common datatype in graphics code, holding homogeneous
Glossary_of_computer_graphics
Mathematics lemma in functional analysis
{\displaystyle 0<\alpha <1.} Then there exists a vector u {\displaystyle u} in X {\displaystyle X} of unit norm ‖ u ‖ = 1 {\displaystyle \|u\|=1} such that
Riesz's_lemma
Arc crossing all meridians of longitude at the same angle
to correspond to latitude), and Cartesian unit vectors i, j, and k can be used to write the radius vector r as r ( λ , φ ) = ( cos λ ⋅ cos φ ) i
Rhumb_line
Expressing a plane wave as a combination of spherical waves
{\mathbf {r} }}),} where i is the imaginary unit, k is a real or complex wave vector of length k, r is a position vector of length r, jℓ are spherical Bessel
Plane-wave_expansion
Frame of reference for an orbit
orbit, i.e. the celestial body about which the orbit is centered. The unit vectors p ^ {\displaystyle \mathbf {\hat {p}} } and q ^ {\displaystyle \mathbf
Perifocal_coordinate_system
Estimation of orbits of objects
the resultant measurement contains only direction information, like a unit vector. With radar, relative distance measurements (by timing of the radar echo)
Orbit_determination
Measure of variation in statistics
{\displaystyle x=\sigma z} , where z {\displaystyle z} has unit variance. In the same way, a random vector x {\displaystyle {\boldsymbol {x}}} in several dimensions
Standard_deviation
Mathematical measure of how much a curve or surface deviates from flatness
per unit distance along the curve. Curvature measures the angular rate of change of the direction of the tangent line, or the unit tangent vector, of
Curvature
Real square matrix whose columns and rows are orthogonal unit vectors
the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I − 2vvT suffices. A Householder reflection
Orthogonal_matrix
Length in a vector space
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance
Norm_(mathematics)
Root-finding algorithm
of squares of the vector components. When each component of the vector is divided by that length, the new vector will be a unit vector pointing in the same
Fast_inverse_square_root
Special mathematical functions defined on the surface of a sphere
{\displaystyle \varphi } only, or equivalently of the orientational unit vector r {\displaystyle \mathbf {r} } specified by these angles. In this setting
Spherical_harmonics
Law of classical electromagnetism
{\displaystyle \mathbf {J} } is the current density vector in that volume (in SI in units of A/m2). In terms of unit vector r ^ ′ {\displaystyle \mathbf {{\hat {r}}'}
Biot–Savart_law
Laws describing planetary orbits
point of view, consider the vector to the planet 𝐫=r𝐫̂, where r is the distance to the planet, and 𝐫̂ is a unit vector pointing towards the planet
Kepler's laws of planetary motion
Kepler's_laws_of_planetary_motion
Set of methods for supervised statistical learning
In machine learning, a support vector machine (SVM) or support vector network is a supervised max-margin model with associated learning algorithms that
Support_vector_machine
Differential geometry topic
function that maps each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface
Gauss_map
Influence that can change motion of an object
direction of a force are both important, force is a vector quantity (force vector). The SI unit of force is the newton (N), and force is often represented
Force
Object movement along a circular path
}}_{R}(t)} is the unit vector parallel to the radius vector at time t and pointing away from the origin. It is convenient to introduce the unit vector orthogonal
Circular_motion
Relativistic generalization of Mordehai Milgrom's MOND paradigm
gravitational lensing. The theory is based on the following ingredients: A unit vector field; A dynamical scalar field; A nondynamical scalar field; A matter
Tensor–vector–scalar_gravity
Technique to reduce dimensionality of points in Euclidean space
{\displaystyle O(ckN)} . A unit vector can be orthogonally projected to a random subspace. Let u {\displaystyle u} be the original unit vector, and let v {\displaystyle
Random_projection
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
Formula in general relativity
_{ij}\right)\right]d^{3}r} where n {\displaystyle \mathbf {n} } is a unit vector in the direction of the observer, r n ≡ r ⋅ n {\displaystyle r_{n}\equiv
Quadrupole_formula
Mathematics term
If ε > 0 and K is a compact subset of G, then a unit vector ξ in H is called an (ε, K)-invariant vector if ∀ g ∈ K : ‖ π ( g ) ξ − ξ ‖ < ε . {\displaystyle
Kazhdan's_property_(T)
computer algorithms. Geometrically, the n-vector for a given position on an ellipsoid is the outward-pointing unit vector that is normal in that position to
N-vector
Topics referred to by the same term
sphere with a radius of length 1 Unit square, a square with sides of length 1 Unit vector, a vector with length equal to 1 Unit type, a type allowing only one
Unit
Derivative of a function with multiple variables
ai+1…,an) −f(a1,…,ai,…,an)h=limh→0f(𝐚+h𝐞i)−f(𝐚)h where 𝐞𝐢 is the unit vector of i-th variable xi. In fact, the last equality shows that the partial
Partial_derivative
Computational tool
The standard unit vector bases of c0, and of ℓp for 1 ≤ p < ∞, are monotone Schauder bases. In this unit vector basis {bn}, the vector bn in V = c0 or
Schauder_basis
Algebraic object with geometric applications
of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There
Tensor
Coordinates comprising a distance and two angles
the local orthogonal unit vectors in the directions of increasing r, θ, and φ, respectively, and x̂, ŷ, and ẑ are the unit vectors in Cartesian coordinates
Spherical_coordinate_system
Geometry calculation
{\displaystyle {\hat {a}}=(a_{x},a_{y},a_{z})} be the cylinder axis unit vector, cylinder radius r {\displaystyle r} , and height (or axis length) h
Line-cylinder_intersection
Statistical model to calculate the value of multiple quantities as they change over time
Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type
Vector_autoregression
Vector representing lattice distortion due to dislocations in a crystal
In materials science, the Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted as b, that represents the magnitude and direction
Burgers_vector
Fundamental physical law of electromagnetism
_{i}} is the vector from its position to r {\displaystyle \mathbf {r} } and r ^ i {\textstyle {\hat {\mathbf {r} }}_{i}} is the unit vector in the direction
Coulomb's_law
Pyramid vector quantization (PVQ) is a method used in audio and video codecs to quantize and transmit unit vectors, i.e. vectors whose magnitudes are known
Pyramid_vector_quantization
Mathematics visualization
denotes a dot product, b ^ {\displaystyle {\hat {\mathbf {b} }}} is the unit vector in the direction of b , {\displaystyle \mathbf {b} ,} ‖ a ‖ {\displaystyle
Scalar_projection
Non-tensorial representation of the spin group
physics, spinors (pronounced "spinner"; /spɪnər/) are elements of a complex vector space that can be associated with Euclidean space. Spinors can be thought
Spinor
Central object in linear algebra; mapping vectors to vectors
{\displaystyle \mathbb {R} ^{m}} and x {\displaystyle \mathbf {x} } is a column vector with n {\displaystyle n} entries, then there exists an m × n {\displaystyle
Transformation_matrix
is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations. Vector algebra uses all dimensions
Comparison of vector algebra and geometric algebra
Comparison_of_vector_algebra_and_geometric_algebra
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UNIT VECTOR
UNIT VECTOR
UNIT VECTOR
UNIT VECTOR
UNIT VECTOR
UNIT VECTOR
UNIT VECTOR
UNIT VECTOR
UNIT VECTOR
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