"In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces."
Fundamental principles of vectors and vector operations.
What are vectors?: Definition and basic properties of vectors.
Vector addition and subtraction: Methods for adding and subtracting vectors, including graphical and algebraic approaches.
Scalar multiplication: How to multiply a vector by a scalar and its properties.
Vector components: How to find the components of a vector in different coordinate systems.
Dot product: Interpretation and properties of the dot product, including its geometric meaning.
Cross product: Interpretation and properties of the cross product, including its geometric meaning.
Vector equations of lines and planes: How to write equations of lines and planes in vector form.
Vector projections: How to find the projection of one vector onto another.
Vector spaces: The properties of vector spaces, including linear independence, basis, and dimension.
Linear transformations: How to represent linear transformations using matrices and vectors.
Eigenvalues and eigenvectors: Definitions and applications of eigenvalues and eigenvectors in linear algebra.
Vector calculus: The application of vectors in calculus, including vector fields, line integrals, and surface integrals.
Applications of vectors: Vectors in physics, engineering, and other fields, including forces, velocities, and accelerations.
Geometric applications: Applications of vectors in geometry, including areas, volumes, and angles.
Vector analysis: Advanced topics in vector analysis, such as vector calculus, differential equations, and vector analysis in higher dimensions.
Column Vector: A column vector is a single column of a matrix. It is represented as [a1;a2;...;an] and is typically used to represent points or vectors in a coordinate system.
Row Vector: A row vector is a single row of a matrix. It is represented as [a1 a2 ... an] and is typically used to represent homogeneous coordinate systems.
Unit Vector: A unit vector is a vector with a length of 1. It is used to represent direction in a coordinate system and is denoted by "hat" symbol above the vector.
Zero Vector: A zero vector is a vector with all its components as 0. It has no length and no direction, and is used to represent the absence of a vector.
Position Vector: A position vector is a vector that represents the position of a point in a coordinate system. It is usually represented as OP, where O is the origin and P is the point.
Displacement Vector: A displacement vector is a vector that represents the displacement or distance between two points in a coordinate system.
Polar Vector: A polar vector is a vector that has a magnitude and direction. It is often represented by an arrow pointing in the direction of its direction with a magnitude represented by the length of the arrow.
Cartesian Vector: A Cartesian vector is a vector that has three components that represent its x, y, and z coordinates in a three-dimensional coordinate system.
Position Vector of a line: A position vector of a line is a vector that represents the position of a line in a coordinate system.
Normal Vector: A normal vector is a vector that is perpendicular to a plane or a line.
Tangent Vector: A tangent vector is a vector that is parallel to a curve or a surface at a given point.
Bi-vector: A bi-vector is a vector that represents the area enclosed by two vectors in a plane. It has direction perpendicular to the plane.
Dual Vector: A dual vector is a vector that is defined by a linear functional acting on a vector space.
Co-vector: A co-vector is a vector that is defined by a linear functional acting on a dual vector space.
Tangent Space Vector: A tangent space vector is a vector that resides in the tangent space of a manifold.
Covariant Vector: A covariant vector is a vector that transforms according to the rules of tensor calculus.
Contravariant Vector: A contravariant vector is a vector that transforms inversely to the rules of tensor calculus.
Inertial Vector: An inertial vector is a vector that is independent of the reference frame in which it is observed.
Relative Vector: A relative vector is a vector that is dependent on the reference frame in which it is observed.
Directed Vector: A directed vector is a vector that has both magnitude and direction. It is used to denote motion or force in a specific direction.
"Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces, and velocity."
"Such quantities are represented by geometric vectors in the same way as distances, masses, and time are represented by real numbers."
"The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length."
"Both geometric vectors and tuples can be added and scaled."
"These vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors."
"A vector space formed by geometric vectors is called a Euclidean vector space."
"A vector space formed by tuples is called a coordinate vector space."
"Many vector spaces are considered in mathematics, such as extension field, polynomial rings, algebras, and function spaces."
"The term vector is generally not used for elements of these vector spaces..."
"...and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces..."
"a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar)..."
"...quantities that have both a magnitude and a direction, such as displacements, forces, and velocity."
"Both geometric vectors and tuples can be added and scaled..."
"...a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors."
"A vector space formed by geometric vectors is called a Euclidean vector space."
"A vector space formed by tuples is called a coordinate vector space."
"...extension field, polynomial rings, algebras, and function spaces."
"The term vector is generally not used for elements of these vector spaces..."
"...and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces."