TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
(or
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...
) and
direction Direction may refer to: *Relative direction, for instance left, right, forward, backwards, up, and down ** Anatomical terms of location for those used in anatomy *Cardinal direction Mathematics and science *Direction vector, a unit vector that ...
. Vectors can be added to other vectors according to
vector algebraIn mathematics, vector algebra may mean: * Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. * The algebraic operations in vector calculus Vector calculus, or vector anal ...
. A Euclidean vector is frequently represented by a ''
ray Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...
'' (a '' directed line segment''), or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by $\overrightarrow$ . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
from ''A'' to ''B''. Many
algebraic operation In mathematics, a basic algebraic operation is any one of the common Operation (mathematics), operations of arithmetic, which include addition, subtraction, multiplication, Division (mathematics), division, raising to an integer exponentiation, powe ...
s on
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s such as
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

, and
negation In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...
have close analogues for vectors, operations which obey the familiar algebraic laws of
commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

,
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
, and
distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. These operations and associated laws qualify
Euclidean Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. It is the name of: Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry a ...
vectors as an example of the more generalized concept of vectors defined simply as elements of a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. Vectors play an important role in
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

: the
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

and
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

of a moving object and the
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

s acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the
coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include
pseudovector In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

s and
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s.

# History

The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions to its development.Michael J. Crowe,
A History of Vector Analysis ''A History of Vector Analysis'' (1967) is a book on the history of vector analysis Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Eucl ...
In 1835,
Giusto Bellavitis Giusto Bellavitis (22 November 1803 – 6 November 1880) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...

abstracted the basic idea when he established the concept of equipollence. Working in a Euclidean plane, he made equipollent any pair of
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
line segments of the same length and orientation. Essentially, he realized an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane. The term ''vector'' was introduced by
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin, Trinity College Dublin, and Dunsink Observatory#Directors, Royal Astronomer ...
as part of a
quaternion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, which is a sum of a
Real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
(also called ''scalar'') and a 3-dimensional ''vector''. Like Bellavitis, Hamilton viewed vectors as representative of
classes Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
of equipollent directed segments. As
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s use an
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad area ...
to complement the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, Hamilton considered the vector to be the ''imaginary part'' of a quaternion: Several other mathematicians developed vector-like systems in the middle of the nineteenth century, including
Augustin Cauchy Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

,
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...
, August Möbius, Comte de Saint-Venant, and Matthew O'Brien. Grassmann's 1840 work ''Theorie der Ebbe und Flut'' (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s.
Peter Guthrie Tait Peter Guthrie Tait FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently dist ...

carried the quaternion standard after Hamilton. His 1867 ''Elementary Treatise of Quaternions'' included extensive treatment of the nabla or
del operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional E ...

∇. In 1878, ''
Elements of Dynamic ''Elements of Dynamic'' is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positive ...
William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...

. Clifford simplified the quaternion study by isolating the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
and
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of two vectors from the complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth.
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in tr ...

, who was exposed to quaternions through
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classica ...

's ''Treatise on Electricity and Magnetism'', separated off their vector part for independent treatment. The first half of Gibbs's ''Elements of Vector Analysis'', published in 1881, presents what is essentially the modern system of vector analysis. In 1901,
Edwin Bidwell Wilson Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
published ''
Vector Analysis Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
'', adapted from Gibb's lectures, which banished any mention of quaternions in the development of vector calculus.

# Overview

In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, a vector is typically regarded as a geometric entity characterized by a
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
and a direction. It is formally defined as a directed
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

, or arrow, in a
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. In
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
, a vector is defined more generally as any element of a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of vectors, as they are elements of a special kind of vector space called
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. This article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric, spatial, or Euclidean vectors. Being an arrow, a Euclidean vector possesses a definite ''initial point'' and ''terminal point''. A vector with fixed initial and terminal point is called a bound vector. When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector. Thus two arrows $\stackrel$ and $\stackrel$ in space represent the same free vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ''ABB′A′'' is a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

. If the Euclidean space is equipped with a choice of
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * , a Wolverine comic book mini-series published by Marvel Comics in 2002 * , a 1999 ''Buffy the Vampire Slayer'' comic book series * , a major ''Judge Dred ...
, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. The term ''vector'' also has generalizations to higher dimensions, and to more formal approaches with much wider applications.

## Further information

In classical
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a sma ...
(i.e.,
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
), vectors were introduced (during the 19th century) as
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
es under equipollence, of
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s of points; two pairs and being equipollent if the points , in this order, form a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

. Such an equivalence class is called a ''vector'', more precisely, a Euclidean vector. The equivalence class of is often denoted $\overrightarrow.$ A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

) and same direction (e.g., the direction from to ). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
s, which have no direction. For example,
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

,
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

s and
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

are represented by vectors. In modern geometry, Euclidean spaces are often defined from
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
. More precisely, a Euclidean space is defined as a set to which is associated an
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of finite dimension over the reals $\overrightarrow,$ and a
group action In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of the
additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
of $\overrightarrow,$ which is
free Free may refer to: Concept * Freedom, having the ability to act or change without constraint * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over their actions and decis ...
and
transitive Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark arg ...
(See
Affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
for details of this construction). The elements of $\overrightarrow$ are called
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
. It has been proven that the two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations. Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of $\mathbb R^n$ equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, $\mathbb R^n$ is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product. The Euclidean space $\mathbb R^n$ is often presented as ''the'' Euclidean space of dimension . This is motivated by the fact that every Euclidean space of dimension is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to the Euclidean space $\mathbb R^n.$ More precisely, given such a Euclidean space, one may choose any point as an
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
. By
Gram–Schmidt process In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, one may also find an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

of any point of the space, as the coordinates on this basis of the vector $\overrightarrow.$ These choices define an isomorphism of the given Euclidean space onto $\mathbb R^n,$ by mapping any point to the -tuple of its Cartesian coordinates, and every vector to its
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...
.

## Examples in one dimension

Since the physicist's concept of
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...
has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force ''F'' of 15
newtons The newton (symbol: N) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...
. If the positive
axis Axis may refer to: Politics *Axis of evil The phrase "axis of evil" was first used by U.S. President George W. Bush in his State of the Union address on January 29, 2002, less than five months after the 9/11 attacks, and often repeated t ...
is also directed rightward, then ''F'' is represented by the vector 15 N, and if positive points leftward, then the vector for ''F'' is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ''s'' of 4
meters The metre (British English, Commonwealth spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek noun , "measure", and ...
would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless.

## In physics and engineering

Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to the rules of vector addition. An example is
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

, the magnitude of which is
speed In everyday use and in kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

. For example, the velocity ''5 meters per second upward'' could be represented by the vector (0, 5) (in 2 dimensions with the positive ''y''-axis as 'up'). Another quantity represented by a vector is
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

, since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement,
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
, linear acceleration,
angular acceleration In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
,
linear momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum ( pl. momenta) is the product of the mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what ...
, and
angular momentum In , angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of . It is an important quantity in physics because it is a —the total angular momentum of a closed system remains constant. In three , the ...

. Other physical vectors, such as the
electric Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnet ...

and
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

, are represented as a system of vectors at each point of a physical space; that is, a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ...

. Examples of quantities that have magnitude and direction, but fail to follow the rules of vector addition, are angular displacement and electric current. Consequently, these are not vectors.

## In Cartesian space

In the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
, a bound vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the points and in space determine the bound vector $\overrightarrow$ pointing from the point on the ''x''-axis to the point on the ''y''-axis. In Cartesian coordinates, a free vector may be thought of in terms of a corresponding bound vector, in this sense, whose initial point has the coordinates of the origin . It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive ''x''-axis. This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector $(1, 2, 3) + (-2, 0, 4) = (1-2, 2+0, 3+4) = (-1, 2, 7)\,.$

## Euclidean and affine vectors

In the geometrical and physical settings, it is sometimes possible to associate, in a natural way, a ''length'' or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an ''angle'' between two vectors. If the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives a convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of a vector by itself). In three dimensions, it is further possible to define the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

, which supplies an algebraic characterization of the
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

and
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
in space of the
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

defined by two vectors (used as sides of the parallelogram). In any dimension (and, in particular, higher dimensions), it's possible to define the
exterior product In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
, which (among other things) supplies an algebraic characterization of the area and orientation in space of the ''n''-dimensional parallelotope defined by ''n'' vectors. In a
pseudo-Euclidean spaceIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, a vector's squared length can be positive, negative, or zero. An important example is
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclid ...
(which is important to our understanding of
special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
). However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...

, where many quantities of interest can be considered vectors in a space with no notion of length or angle.Thermodynamics and Differential Forms
/ref>

## Generalizations

In physics, as well as mathematics, a vector is often identified with a
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of components, or list of numbers, that act as scalar coefficients for a set of
basis vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s. When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but the basis has, so the components of the vector must change to compensate. The vector is called ''covariant'' or ''contravariant'', depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

. If you change units (a special case of a change of basis) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm—a contravariant change in numerical value. In contrast, a gradient of 1 /m becomes 0.001 K/mm—a covariant change in value (for more, see
covariance and contravariance of vectors In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
).
Tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

s are another type of quantity that behave in this way; a vector is one type of
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. In pure
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a vector is any element of a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
over some
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
and is often represented as a
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...
. The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction".

# Representations

Vectors are usually denoted in
lowercase Letter case is the distinction between the letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word that indicates, signifies, or i ...
boldface, as in $\mathbf$, $\mathbf$ and $\mathbf$, or in lowercase italic boldface, as in ''a''. (
Uppercase Letter case is the distinction between the letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word that indicates, signifies, or ...
letters are typically used to represent
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
.) Other conventions include $\vec$ or ''a'', especially in handwriting. Alternatively, some use a
tilde The tilde (
in the American Heritage dictionary
), or , is a

(~) or a wavy underline drawn beneath the symbol, e.g. $\underseta$, which is a convention for indicating boldface type. If the vector represents a directed
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

or
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
from a point ''A'' to a point ''B'' (see figure), it can also be denoted as $\stackrel$ or ''AB''. In
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ...

literature, it was especially common to represent vectors with small
fraktur Fraktur () is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand. The blackletter lines are broken up; that is, their forms contain many angles when compared to the curves of the Antiqua ...

letters such as $\mathfrak$. Vectors are usually shown in graphs or other diagrams as arrows (directed
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

s), as illustrated in the figure. Here, the point ''A'' is called the ''origin'', ''tail'', ''base'', or ''initial point'', and the point ''B'' is called the ''head'', ''tip'', ''endpoint'', ''terminal point'' or ''final point''. The length of the arrow is proportional to the vector's
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
, while the direction in which the arrow points indicates the vector's direction. On a two-dimensional diagram, a vector
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

to the
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip of an
arrow An arrow is a fin-stabilized projectile launched by a bow and arrow, bow. A typical arrow usually consists of a long, stiff, straight ''shaft'' with a weighty (and usually sharp and pointed) ''arrowhead'' attached to the front end, multiple fi ...
head on and viewing the flights of an arrow from the back. In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an ''n''-dimensional Euclidean space can be represented as
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...
s in a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
. The endpoint of a vector can be identified with an ordered list of ''n'' real numbers (''n''-
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
). These numbers are the
coordinates In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

of the endpoint of the vector, with respect to a given
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
, and are typically called the scalar components (or scalar projections) of the vector on the axes of the coordinate system. As an example in two dimensions (see figure), the vector from the origin ''O'' = (0, 0) to the point ''A'' = (2, 3) is simply written as $\mathbf = (2,3).$ The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation $\overrightarrow$ is usually deemed not necessary (and is indeed rarely used). In ''three dimensional'' Euclidean space (or ), vectors are identified with triples of scalar components: $\mathbf = (a_1, a_2, a_3).$ also written, $\mathbf = (a_x, a_y, a_z).$ This can be generalised to ''n-dimensional'' Euclidean space (or ). $\mathbf = (a_1, a_2, a_3, \cdots, a_, a_n).$ These numbers are often arranged into a
column vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and ...
or
row vector In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and th ...
, particularly when dealing with
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
, as follows: Another way to represent a vector in ''n''-dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them: $_1 = (1,0,0),\ _2 = (0,1,0),\ _3 = (0,0,1).$ These have the intuitive interpretation as vectors of unit length pointing up the ''x''-, ''y''-, and ''z''-axis of a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
, respectively. In terms of these, any vector a in can be expressed in the form: $\mathbf = (a_1,a_2,a_3) = a_1(1,0,0) + a_2(0,1,0) + a_3(0,0,1), \$ or $\mathbf = \mathbf_1 + \mathbf_2 + \mathbf_3 = a_1_1 + a_2_2 + a_3_3,$ where a1, a2, a3 are called the vector components (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes ''x'', ''y'', and ''z'' (see figure), while ''a''1, ''a''2, ''a''3 are the respective scalar components (or scalar projections). In introductory physics textbooks, the standard basis vectors are often denoted $\mathbf,\mathbf,\mathbf$ instead (or $\mathbf, \mathbf, \mathbf$, in which the hat symbol ^ typically denotes unit vectors). In this case, the scalar and vector components are denoted respectively ''ax'', ''ay'', ''az'', and a''x'', a''y'', a''z'' (note the difference in boldface). Thus, $\mathbf = \mathbf_x + \mathbf_y + \mathbf_z = a_x + a_y + a_z.$ The notation e''i'' is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering.

## Decomposition or resolution

As explained #Representations, above, a vector is often described by a set of vector components that #Addition and subtraction, add up to form the given vector. Typically, these components are the Vector projection, projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be ''decomposed'' or ''resolved with respect to'' that set. The decomposition or resolution of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as $\mathbf, \mathbf, \mathbf$ as a Basis (linear algebra), basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including the unit vectors of a cylindrical coordinate system ($\boldsymbol, \boldsymbol, \mathbf$) or spherical coordinate system ($\mathbf, \boldsymbol, \boldsymbol$). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively. The choice of a basis does not affect the properties of a vector or its behaviour under transformations. A vector can also be broken up with respect to "non-fixed" basis vectors that change their
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building design ...
as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively ''normal'', and ''tangent'' to a surface (see figure). Moreover, the ''radial'' and ''tangential components'' of a vector relate to the ''radius of rotation'' of an object. The former is Parallel (geometry), parallel to the radius and the latter is Perpendicular, orthogonal to it. In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a ''global'' coordinate system, or inertial reference frame).

# Basic properties

The following section uses the
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
with basis vectors $_1 = (1,0,0),\ _2 = (0,1,0),\ _3 = (0,0,1)$ and assumes that all vectors have the origin as a common base point. A vector a will be written as $= a_1_1 + a_2_2 + a_3_3.$

## Equality

Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors $= a_1_1 + a_2_2 + a_3_3$ and $= b_1_1 + b_2_2 + b_3_3$ are equal if $a_1 = b_1,\quad a_2=b_2,\quad a_3=b_3.\,$

## Opposite, parallel, and antiparallel vectors

Two vectors are opposite if they have the same magnitude but opposite direction. So two vectors $= a_1_1 + a_2_2 + a_3_3$ and $= b_1_1 + b_2_2 + b_3_3$ are opposite if $a_1 = -b_1,\quad a_2=-b_2,\quad a_3=-b_3.\,$ Two vectors are parallel if they have the same direction but not necessarily the same magnitude, or antiparallel if they have opposite direction but not necessarily the same magnitude.

Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. The sum of a and b is $\mathbf+\mathbf =(a_1+b_1)\mathbf_1 +(a_2+b_2)\mathbf_2 +(a_3+b_3)\mathbf_3.$ The addition may be represented graphically by placing the tail of the arrow b at the head of the arrow a, and then drawing an arrow from the tail of a to the head of b. The new arrow drawn represents the vector a + b, as illustrated below: This addition method is sometimes called the ''parallelogram rule'' because a and b form the sides of a
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

and a + b is one of the diagonals. If a and b are bound vectors that have the same base point, this point will also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). The difference of a and b is $\mathbf-\mathbf =(a_1-b_1)\mathbf_1 +(a_2-b_2)\mathbf_2 +(a_3-b_3)\mathbf_3.$ Subtraction of two vectors can be geometrically illustrated as follows: to subtract b from a, place the tails of a and b at the same point, and then draw an arrow from the head of b to the head of a. This new arrow represents the vector (-b) + a, with (-b) being the opposite of b, see drawing. And (-b) + a = a − b.

## Scalar multiplication

A vector may also be multiplied, or re-''scaled'', by a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
''r''. In the context of vector analysis, conventional vector algebra, these real numbers are often called scalars (from ''scale'') to distinguish them from vectors. The operation of multiplying a vector by a scalar is called ''scalar multiplication''. The resulting vector is $r\mathbf=(ra_1)\mathbf_1 +(ra_2)\mathbf_2 +(ra_3)\mathbf_3.$ Intuitively, multiplying by a scalar ''r'' stretches a vector out by a factor of ''r''. Geometrically, this can be visualized (at least in the case when ''r'' is an integer) as placing ''r'' copies of the vector in a line where the endpoint of one vector is the initial point of the next vector. If ''r'' is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (''r'' = −1 and ''r'' = 2) are given below: Scalar multiplication is Distributivity, distributive over vector addition in the following sense: ''r''(a + b) = ''r''a + ''r''b for all vectors a and b and all scalars ''r''. One can also show that a − b = a + (−1)b.

## Length

The ''length'' or ''
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
'' or ''Norm (mathematics), norm'' of the vector a is denoted by ‖a‖ or, less commonly, , a, , which is not to be confused with the absolute value (a scalar "norm"). The length of the vector a can be computed with the Euclidean norm, $\left\, \mathbf\right\, =\sqrt,$ which is a consequence of the Pythagorean theorem since the basis vectors e1, e2, e3 are orthogonal unit vectors. This happens to be equal to the square root of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
, discussed below, of the vector with itself: $\left\, \mathbf\right\, =\sqrt.$

### Unit vector

A ''unit vector'' is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. This is known as ''normalizing'' a vector. A unit vector is often indicated with a hat as in â. To normalize a vector , scale the vector by the reciprocal of its length ‖a‖. That is: $\mathbf = \frac = \frac\mathbf_1 + \frac\mathbf_2 + \frac\mathbf_3$

### Zero vector

The ''zero vector'' is the vector with length zero. Written out in coordinates, the vector is , and it is commonly denoted $\vec$, 0, or simply 0. Unlike any other vector, it has an arbitrary or indeterminate direction, and cannot be normalized (that is, there is no unit vector that is a multiple of the zero vector). The sum of the zero vector with any vector a is a (that is, ).

## Dot product

The ''dot product'' of two vectors a and b (sometimes called the ''inner product space, inner product'', or, since its result is a scalar, the ''scalar product'') is denoted by a ∙ b, and is defined as: $\mathbf\cdot\mathbf =\left\, \mathbf\right\, \left\, \mathbf\right\, \cos\theta,$ where ''θ'' is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point, and then the length of a is multiplied with the length of the component of b that points in the same direction as a. The dot product can also be defined as the sum of the products of the components of each vector as $\mathbf \cdot \mathbf = a_1 b_1 + a_2 b_2 + a_3 b_3.$

## Cross product

The ''cross product'' (also called the ''vector product'' or ''outer product'') is only meaningful in three or Seven-dimensional cross product, seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as $\mathbf\times\mathbf =\left\, \mathbf\right\, \left\, \mathbf\right\, \sin(\theta)\,\mathbf$ where ''θ'' is the measure of the angle between a and b, and n is a unit vector
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

to both a and b which completes a Right-hand rule, right-handed system. The right-handedness constraint is necessary because there exist ''two'' unit vectors that are perpendicular to both a and b, namely, n and (−n). The cross product a × b is defined so that a, b, and a × b also becomes a right-handed system (although a and b are not necessarily orthogonal). This is the right-hand rule. The length of a × b can be interpreted as the area of the parallelogram having a and b as sides. The cross product can be written as $\times = (a_2 b_3 - a_3 b_2) _1 + (a_3 b_1 - a_1 b_3) _2 + (a_1 b_2 - a_2 b_1) _3.$ For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a
pseudovector In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

instead of a vector (see below).

## Scalar triple product

The ''scalar triple product'' (also called the ''box product'' or ''mixed triple product'') is not really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is sometimes denoted by (a b c) and defined as: $(\mathbf\ \mathbf\ \mathbf) =\mathbf\cdot(\mathbf\times\mathbf).$ It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linear independence, linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are right-handed. In components (''with respect to a right-handed orthonormal basis''), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the determinant of the 3-by-3 Matrix (mathematics), matrix having the three vectors as rows $(\mathbf\ \mathbf\ \mathbf)=\left, \begin a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end\$ The scalar triple product is linear in all three entries and anti-symmetric in the following sense: $(\mathbf\ \mathbf\ \mathbf) = (\mathbf\ \mathbf\ \mathbf) = (\mathbf\ \mathbf\ \mathbf)= -(\mathbf\ \mathbf\ \mathbf) = -(\mathbf\ \mathbf\ \mathbf) = -(\mathbf\ \mathbf\ \mathbf).$

## Conversion between multiple Cartesian bases

All examples thus far have dealt with vectors expressed in terms of the same basis, namely, the ''e'' basis . However, a vector can be expressed in terms of any number of different bases that are not necessarily aligned with each other, and still remain the same vector. In the ''e'' basis, a vector a is expressed, by definition, as $\mathbf = p\mathbf_1 + q\mathbf_2 + r\mathbf_3.$ The scalar components in the ''e'' basis are, by definition, $\begin p &= \mathbf\cdot\mathbf_1, \\ q &= \mathbf\cdot\mathbf_2, \\ r &= \mathbf\cdot\mathbf_3. \end$ In another orthonormal basis ''n'' = that is not necessarily aligned with ''e'', the vector a is expressed as $\mathbf = u\mathbf_1 + v\mathbf_2 + w\mathbf_3$ and the scalar components in the ''n'' basis are, by definition, $\begin u &= \mathbf\cdot\mathbf_1, \\ v &= \mathbf\cdot\mathbf_2, \\ w &= \mathbf\cdot\mathbf_3. \end$ The values of ''p'', ''q'', ''r'', and ''u'', ''v'', ''w'' relate to the unit vectors in such a way that the resulting vector sum is exactly the same physical vector a in both cases. It is common to encounter vectors known in terms of different bases (for example, one basis fixed to the Earth and a second basis fixed to a moving vehicle). In such a case it is necessary to develop a method to convert between bases so the basic vector operations such as addition and subtraction can be performed. One way to express ''u'', ''v'', ''w'' in terms of ''p'', ''q'', ''r'' is to use column matrices along with a direction cosine matrix containing the information that relates the two bases. Such an expression can be formed by substitution of the above equations to form $\begin u &= (p\mathbf_1 + q\mathbf_2 + r\mathbf_3)\cdot\mathbf_1, \\ v &= (p\mathbf_1 + q\mathbf_2 + r\mathbf_3)\cdot\mathbf_2, \\ w &= (p\mathbf_1 + q\mathbf_2 + r\mathbf_3)\cdot\mathbf_3. \end$ Distributing the dot-multiplication gives $\begin u &= p\mathbf_1\cdot\mathbf_1 + q\mathbf_2\cdot\mathbf_1 + r\mathbf_3\cdot\mathbf_1, \\ v &= p\mathbf_1\cdot\mathbf_2 + q\mathbf_2\cdot\mathbf_2 + r\mathbf_3\cdot\mathbf_2, \\ w &= p\mathbf_1\cdot\mathbf_3 + q\mathbf_2\cdot\mathbf_3 + r\mathbf_3\cdot\mathbf_3. \end$ Replacing each dot product with a unique scalar gives $\begin u &= c_p + c_q + c_r, \\ v &= c_p + c_q + c_r, \\ w &= c_p + c_q + c_r, \end$ and these equations can be expressed as the single matrix equation $\begin u \\ v \\ w \\ \end = \begin c_ & c_ & c_ \\ c_ & c_ & c_ \\ c_ & c_ & c_ \end \begin p \\ q \\ r \end.$ This matrix equation relates the scalar components of a in the ''n'' basis (''u'',''v'', and ''w'') with those in the ''e'' basis (''p'', ''q'', and ''r''). Each matrix element ''c''''jk'' is the Direction cosine#Cartesian coordinates, direction cosine relating n''j'' to e''k''. The term ''direction cosine'' refers to the cosine of the angle between two unit vectors, which is also equal to their #Dot product, dot product. Therefore, $\begin c_ &= \mathbf_1\cdot\mathbf_1 \\ c_ &= \mathbf_1\cdot\mathbf_2 \\ c_ &= \mathbf_1\cdot\mathbf_3 \\ c_ &= \mathbf_2\cdot\mathbf_1 \\ c_ &= \mathbf_2\cdot\mathbf_2 \\ c_ &= \mathbf_2\cdot\mathbf_3 \\ c_ &= \mathbf_3\cdot\mathbf_1 \\ c_ &= \mathbf_3\cdot\mathbf_2 \\ c_ &= \mathbf_3\cdot\mathbf_3 \end$ By referring collectively to e1, e2, e3 as the ''e'' basis and to n1, n2, n3 as the ''n'' basis, the matrix containing all the ''c''''jk'' is known as the "transformation matrix from ''e'' to ''n''", or the "rotation matrix from ''e'' to ''n''" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "direction cosine matrix from ''e'' to ''n''" (because it contains direction cosines). The properties of a rotation matrix are such that its matrix inverse, inverse is equal to its matrix transpose, transpose. This means that the "rotation matrix from ''e'' to ''n''" is the transpose of "rotation matrix from ''n'' to ''e''". The properties of a direction cosine matrix, C are: * the determinant is unity, , C, = 1; * the inverse is equal to the transpose; * the rows and columns are orthogonal unit vectors, therefore their dot products are zero. The advantage of this method is that a direction cosine matrix can usually be obtained independently by using Euler angles or a
quaternion In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

to relate the two vector bases, so the basis conversions can be performed directly, without having to work out all the dot products described above. By applying several matrix multiplications in succession, any vector can be expressed in any basis so long as the set of direction cosines is known relating the successive bases.

## Other dimensions

With the exception of the cross and triple products, the above formulae generalise to two dimensions and higher dimensions. For example, addition generalises to two dimensions as $(a_1_1 + a_2_2)+(b_1_1 + b_2_2) = (a_1+b_1)_1 + (a_2+b_2)_2,$ and in four dimensions as $\begin (a_1_1 + a_2_2 + a_3_3 + a_4_4) &+ (b_1_1 + b_2_2 + b_3_3 + b_4_4) =\\ (a_1+b_1)_1 + (a_2+b_2)_2 &+ (a_3+b_3)_3 + (a_4+b_4)_4. \end$ The cross product does not readily generalise to other dimensions, though the closely related Exterior algebra#Areas in the plane, exterior product does, whose result is a bivector. In two dimensions this is simply a pseudoscalar $(a_1_1 + a_2_2)\wedge(b_1_1 + b_2_2) = (a_1 b_2 - a_2 b_1)\mathbf_1 \mathbf_2.$ A seven-dimensional cross product is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products.

# Physics

Vectors have many uses in physics and other sciences.

## Length and units

In abstract vector spaces, the length of the arrow depends on a Dimensionless number, dimensionless Scale (measurement), scale. If it represents, for example, a force, the "scale" is of Dimensional analysis, physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2 cm, the scales are 1 m:50 N and 1:250 respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.

## Vector-valued functions

Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parameter ''t''. For instance, if r represents the position vector of a particle, then r(''t'') gives a parametric equation, parametric representation of the trajectory of the particle. Vector-valued functions can be derivative, differentiated and integral, integrated by differentiating or integrating the components of the vector, and many of the familiar rules from calculus continue to hold for the derivative and integral of vector-valued functions.

## Position, velocity and acceleration

The position of a point x = (''x''1, ''x''2, ''x''3) in three-dimensional space can be represented as a position vector whose base point is the origin $= x_1 _1 + x_2_2 + x_3_3.$ The position vector has dimensions of length. Given two points x = (''x''1, ''x''2, ''x''3), y = (''y''1, ''y''2, ''y''3) their Displacement (vector), displacement is a vector $-=(y_1-x_1)_1 + (y_2-x_2)_2 + (y_3-x_3)_3.$ which specifies the position of ''y'' relative to ''x''. The length of this vector gives the straight-line distance from ''x'' to ''y''. Displacement has the dimensions of length. The
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

v of a point or particle is a vector, its length gives the
speed In everyday use and in kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

. For constant velocity the position at time ''t'' will be $_t= t + _0,$ where x0 is the position at time ''t'' = 0. Velocity is the #Ordinary derivative, time derivative of position. Its dimensions are length/time. Acceleration a of a point is vector which is the #Ordinary derivative, time derivative of velocity. Its dimensions are length/time2.

## Force, energy, work

Force is a vector with dimensions of mass×length/time2 and Newton's second law is the scalar multiplication $= m$ Work is the dot product of
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

and
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
$E = \cdot (_2 - _1).$

# Vectors, pseudovectors, and transformations

An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a coordinate system, coordinate transformation. A ''contravariant vector'' is required to have components that "transform opposite to the basis" under changes of Basis (linear algebra), basis. The vector itself does not change when the basis is transformed; instead, the components of the vector make a change that cancels the change in the basis. In other words, if the reference axes (and the basis derived from it) were rotated in one direction, the component representation of the vector would rotate in the opposite way to generate the same final vector. Similarly, if the reference axes were stretched in one direction, the components of the vector would reduce in an exactly compensating way. Mathematically, if the basis undergoes a transformation described by an invertible matrix ''M'', so that a coordinate vector x is transformed to , then a contravariant vector v must be similarly transformed via . This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, if ''v'' consists of the ''x'', ''y'', and ''z''-components of
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

, then ''v'' is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an abstract vector space, vector, but this vector would not be contravariant, since rotating the box does not change the box's length, width, and height. Examples of contravariant vectors include
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
,
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

, electric field, momentum,
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

, and
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

. In the language of differential geometry, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a ''contravariant vector'' to be a
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

of Covariance and contravariance of vectors, contravariant rank one. Alternatively, a contravariant vector is defined to be a tangent space, tangent vector, and the rules for transforming a contravariant vector follow from the chain rule. Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the ''orientation (space), orientation'' of space. A vector which gains a minus sign when the orientation of space changes is called a ''
pseudovector In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...

'' or an ''axial vector''. Ordinary vectors are sometimes called ''true vectors'' or ''polar vectors'' to distinguish them from pseudovectors. Pseudovectors occur most frequently as the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of two ordinary vectors. One example of a pseudovector is angular velocity. Driving in a car, and looking forward, each of the wheels has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, the ''reflection'' of this angular velocity vector points to the right, but the angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors include
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ...

, torque, or more generally any cross product of two (true) vectors. This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties. See parity (physics).

*
Affine space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, which distinguishes between vectors and Point (geometry), points * Array data structure or Vector (Computer Science) * Banach space * Clifford algebra * Complex number * Coordinate system * Covariance and contravariance of vectors * Four-vector, a non-Euclidean vector in Minkowski space (i.e. four-dimensional spacetime), important in theory of relativity, relativity * Function space * Grassmann's ''Ausdehnungslehre'' * Hilbert space * Normal vector * Null vector * Position (geometry) * Pseudovector * Quaternion * Tangential and normal components (of a vector) *
Tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

* Unit vector * Vector bundle * Vector calculus * Vector notation * Vector-valued function

# References

## Mathematical treatments

* * *. *. *. *. * *

* *