Examples of vectors in nature are velocity, momentum, force, electromagnetic fields and weight. A quantity or phenomenon that exhibits magnitude only, with no specific direction, is called a scalar. Examples of scalars include speed, mass, electrical resistance and hard drive storage capacity.

Vectors are typically represented by an arrow with a beginning, or tail, and an end, or head, that is usually represented by an arrowhead. Vectors delineate the movement from point A to point B and can be defined as an entity with a designation, such as vector a.

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In mathematics, the Cartesian coordinate system depicts vectors using a number pair as shown in Figure 1. In this example, the vector is a directed line segment defined as (0,0), (7,7) using its numbered pairs. Vectors and scalars can be used in mathematical processes and vector operations, such as vector addition, vector subtraction and vector multiplication.

Vectors can be depicted graphically in two or three dimensions. The magnitude of the vector is shown as the length of a line segment. The direction of the vector is shown by the orientation of the line segment and by an arrowhead at one end.

When creating images in vector graphics, the graphics file is a sequence of vector statements describing a series of points that connect to form the image. Examples of vector graphics software include Adobe Illustrator and CorelDraw.

raster graphics map individual bits -- each of which has its own qualities, such as color -- into an image via components called pixels, or picture elements. Raster images typically have a fixed number of pixels and are less scalable than vector images. As an image gets larger, individual pixels can become visible, resulting in the image being not as sharp or high quality as an equivalent vector image.

With raster images, computers must store each pixel, rather than a series of vector points. This often results in raster files being larger than vector graphics files. Adobe Illustrator and Adobe Photoshop are examples of software that is used to convert raster images into vector graphics files and vice versa.

In cybersecurity, the pathway that a threat actor or hacker uses to deliver their payload, such as a virus or ransomware, to a system or network is called an attack vector. This is how hackers exploit vulnerabilities in a system or network.

An attack vector can also be human, in the case of social engineering, where the perpetrator uses clever communication techniques to mislead users into giving out valuable information such as passwords. Typical attack vectors include malware, email attachments, instant messages and pop-up windows.

Cyber threat detection and mitigation systems include firewalls, intrusion detection systems, intrusion prevention systems and antivirus software. Numerous products and services are available to prevent threat actors from using attack vectors and to address threats.

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.

Both geometric vectors and tuples can be added and scaled, and 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, and a vector space formed by tuples is called a coordinate vector space.

Many vector spaces are considered in mathematics, such as extension fields, 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 (for example, when discussing general properties of vector spaces).

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".[4] It was first used by 18th century astronomers investigating planetary revolution around the Sun.[5] The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors,[6] operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars: real coordinate space or complex coordinate space.

Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrices, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equations.

Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension.

Calculus serves as a foundational mathematical tool in the realm of vectors, offering a framework for the analysis and manipulation of vector quantities in diverse scientific disciplines, notably physics and engineering. Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential insights into motion within three-dimensional space. Vector calculus extends traditional calculus principles to vector fields, introducing operations like gradient, divergence, and curl, which find applications in physics and engineering contexts. Line integrals, crucial for calculating work along a path within force fields, and surface integrals, employed to determine quantities like flux, illustrate the practical utility of calculus in vector analysis. Volume integrals, essential for computations involving scalar or vector fields over three-dimensional regions, contribute to understanding mass distribution, charge density, and fluid flow rates.[citation needed]

The elements are stored contiguously, which means that elements can be accessed not only through iterators, but also using offsets to regular pointers to elements. This means that a pointer to an element of a vector may be passed to any function that expects a pointer to an element of an array.

The storage of the vector is handled automatically, being expanded as needed. Vectors usually occupy more space than static arrays, because more memory is allocated to handle future growth. This way a vector does not need to reallocate each time an element is inserted, but only when the additional memory is exhausted. The total amount of allocated memory can be queried using capacity() function. Extra memory can be returned to the system via a call to shrink_to_fit()[1].

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