In

force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

, 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,

/ref>

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 ...

over some

__''a''__, especially in handwriting. Alternatively, some use a 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 ...

from a point ''A'' to a point ''B'' (see figure), it can also be denoted as $\backslash stackrel$ or __''AB''__. In Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, respectively. In terms of these, any vector a in can be expressed in the form:
$$\backslash mathbf\; =\; (a\_1,a\_2,a\_3)\; =\; a\_1(1,0,0)\; +\; a\_2(0,1,0)\; +\; a\_3(0,0,1),\; \backslash $$
or
$$\backslash mathbf\; =\; \backslash mathbf\_1\; +\; \backslash mathbf\_2\; +\; \backslash mathbf\_3\; =\; a\_1\_1\; +\; a\_2\_2\; +\; a\_3\_3,$$
where a_{1}, a_{2}, a_{3} are called the _{1}, ''a''_{2}, ''a''_{3} are the respective scalar components (or scalar projections).
In introductory physics textbooks, the standard basis vectors are often denoted $\backslash mathbf,\backslash mathbf,\backslash mathbf$ instead (or $\backslash mathbf,\; \backslash mathbf,\; \backslash mathbf$, in which the hat symbol ^ typically denotes _{x}'', ''a_{y}'', ''a_{z}'', and a_{''x''}, a_{''y''}, a_{''z''} (note the difference in boldface). Thus,
$$\backslash mathbf\; =\; \backslash mathbf\_x\; +\; \backslash mathbf\_y\; +\; \backslash mathbf\_z\; =\; a\_x\; +\; a\_y\; +\; a\_z.$$
The notation e_{''i''} is compatible with the

Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

with basis vectors
$$\_1\; =\; (1,0,0),\backslash \; \_2\; =\; (0,1,0),\backslash \; \_3\; =\; (0,0,1)$$
and assumes that all vectors have the origin as a common base point. A vector a will be written as
$$=\; a\_1\_1\; +\; a\_2\_2\; +\; a\_3\_3.$$

parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

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
$$\backslash mathbf-\backslash mathbf\; =(a\_1-b\_1)\backslash mathbf\_1\; +(a\_2-b\_2)\backslash mathbf\_2\; +(a\_3-b\_3)\backslash mathbf\_3.$$
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.

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 ''_{1}, e_{2}, e_{3} 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'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

, discussed below, of the vector with itself:
$$\backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; =\backslash sqrt.$$

_{''jk''} is the 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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

. Therefore,
$$\backslash begin\; c\_\; \&=\; \backslash mathbf\_1\backslash cdot\backslash mathbf\_1\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_1\backslash cdot\backslash mathbf\_2\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_1\backslash cdot\backslash mathbf\_3\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_2\backslash cdot\backslash mathbf\_1\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_2\backslash cdot\backslash mathbf\_2\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_2\backslash cdot\backslash mathbf\_3\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_3\backslash cdot\backslash mathbf\_1\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_3\backslash cdot\backslash mathbf\_2\; \backslash \backslash \; c\_\; \&=\; \backslash mathbf\_3\backslash cdot\backslash mathbf\_3\; \backslash end$$
By referring collectively to e_{1}, e_{2}, e_{3} as the ''e'' basis and to n_{1}, n_{2}, n_{3} as the ''n'' basis, the matrix containing all the ''c''_{''jk''} is known as the "

_{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 _{1}, ''x''_{2}, ''x''_{3}), y = (''y''_{1}, ''y''_{2}, ''y''_{3}) their 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 ...

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
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

v of a point or particle is a vector, its length gives the _{0} is the position at time ''t'' = 0. Velocity is the ^{2}.

^{2} and force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

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 ...

$$E\; =\; \backslash cdot\; (\_2\; -\; \_1).$$

velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

, 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 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 ...

, velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

, force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

, and

Online vector identities

(

Introducing Vectors

A conceptual introduction (

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, and engineering
Engineering is the use of scientific method, 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 rang ...

, 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. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

) and direction. Vectors can be added to other vectors according to vector algebra In 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, namely the specific additional stru ...

. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by $\backslash overrightarrow$ .
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 ...

from ''A'' to ''B''. Many algebraic operation
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...

s on real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

s such as addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...

, 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 mathematical operations of arithmetic, with the other ones being additi ...

, and negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...

have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

, 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 valid rule of replacement f ...

, and distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...

. 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 ...

.
Vectors play an important role in physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

: the velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

and acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...

of a moving object and the force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

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
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...

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 ...

), 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, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

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 and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...

s and tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

s.
History

The vector concept, as we know it today, is the result of a gradual development over a period of more than 200 years. About a dozen people contributed significantly 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 by Michael J. Crowe, originally published by the University of Notre Dame Press.
As a scholarly treatment of a reformation in technical communication, the text i ...

; see also his on the subject. In 1835, Giusto Bellavitis abstracted the basic idea when he established the concept of equipollence. Working in a Euclidean plane, he made equipollent any pair of parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...

line segments of the same length and orientation. Essentially, he realized an equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...

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, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...

as part of a quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

, which is a sum of a Real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

(also called ''scalar'') and a 3-dimensional ''vector''. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments. As complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s use an imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...

to complement the real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

, 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 Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...

, 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 linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...

, August Möbius
August is the eighth month of the year in the Julian and Gregorian calendars, and the fifth of seven months to have a length of 31 days. Its zodiac sign is Leo and was originally named ''Sextilis'' in Latin because it was the 6th month in ...

, Comte de Saint-Venant
''Comte'' is the French, Catalan and Occitan form of the word 'count' (Latin: ''comes''); ''comté'' is the Gallo-Romance form of the word 'county' (Latin: ''comitatus'').
Comte or Comté may refer to:
* A count in French, from Latin ''comes''
* A ...

, and Matthew O'Brien
Matthew O'Brien (born in 1970) is an American author, journalist, editor and teacher who writes about the seedier side of Las Vegas. His most well-known work is the nonfiction book '' Beneath the Neon'', which documents the homeless population li ...

. 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 (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote wi ...

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) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...

∇. 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 positiv ...

'' was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

and cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

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 t ...

, who was exposed to quaternions through James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

'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, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...

published ''Vector Analysis
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...

'', adapted from Gibb's lectures, which banished any mention of quaternions in the development of vector calculus.
Overview

Inphysics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

and engineering
Engineering is the use of scientific method, 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 rang ...

, 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
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

, or arrow, in a Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. In pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...

, a vector is defined more generally as any element 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 ...

. 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 geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. This particular 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 $\backslash stackrel$ and $\backslash stackrel$ 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, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

. 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
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

, 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 classicalEuclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...

(i.e., synthetic geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compa ...

), vectors were introduced (during the 19th century) as equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

es under equipollence, of ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...

s of points; two pairs and being equipollent if the points , in this order, form a parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

. Such an equivalence class is called a ''vector'', more precisely, a Euclidean vector. The equivalence class of is often denoted $\backslash overrightarrow.$
A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

) 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 ...

s, which have no direction. For example, velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

, force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

s and acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...

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 matrices.
...

. More precisely, a Euclidean space is defined as a set to which is associated an inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...

of finite dimension over the reals $\backslash overrightarrow,$ and a group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

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 structures ...

of $\backslash overrightarrow,$ which is free and transitive (See Affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

for details of this construction). The elements of $\backslash overrightarrow$ 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 ''transl ...

. 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 $\backslash mathbb\; R^n$ equipped with the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, $\backslash mathbb\; R^n$ is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.
The Euclidean space $\backslash mathbb\; R^n$ 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, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

to the Euclidean space $\backslash mathbb\; R^n.$ 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), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

. By Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner produ ...

, one may also find an orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...

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 of any point of the space, as the coordinates on this basis of the vector $\backslash overrightarrow.$ These choices define an isomorphism of the given Euclidean space onto $\backslash mathbb\; R^n,$ by mapping any point to the -tuple of its Cartesian coordinates, and every vector to its coordinate vector.
Examples in one dimension

Since the physicist's concept offorce
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force ''F'' of 15 newtons. If the positive axis
An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to:
Mathematics
* Axis of rotation: see rotation around a fixed axis
* Axis (mathematics), a designator for a Cartesian-coordinat ...

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 spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its p ...

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 isvelocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

, the magnitude of which is speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...

. For instance, 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 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 ...

, linear acceleration, angular acceleration, linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...

, and angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...

. Other physical vectors, such as the electric
Electricity is the set of physical 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 electromagnetism, as described by ...

and magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

, are represented as a system of vectors at each point of a physical space; that is, a vector field. 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 theCartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, 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 $\backslash overrightarrow$ 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
$$(1,\; 2,\; 3)\; +\; (-2,\; 0,\; 4)\; =\; (1-2,\; 2+0,\; 3+4)\; =\; (-1,\; 2,\; 7)\backslash ,.$$
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 thedot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...

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 mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

, which supplies an algebraic characterization of the area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...

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 de ...

in space of the parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

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 mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...

, 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 space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non- degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving
q(x ...

, 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 and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

(which is important to our understanding of special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...

).
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 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 ...

s (for free vectors) and affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

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, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...

, 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 atuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

of components, or list of numbers, that act as scalar coefficients for a set of basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...

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, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...

. 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 K/m becomes 0.001 K/mm—a covariant change in value (for more, see covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation ...

). Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

s are another type of quantity that behave in this way; a vector is one type of tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

.
In pure mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a vector is any element of a 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 grass ...

and is often represented as a coordinate vector. 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 inlowercase
Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...

boldface, as in $\backslash mathbf$, $\backslash mathbf$ and $\backslash mathbf$, or in lowercase italic boldface, as in ''a''. (Uppercase
Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...

letters are typically used to represent matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

.) Other conventions include $\backslash vec$ or tilde
The tilde () or , is a grapheme with several uses. The name of the character came into English from Spanish, which in turn came from the Latin '' titulus'', meaning "title" or "superscription". Its primary use is as a diacritic (accent) in ...

(~) or a wavy underline drawn beneath the symbol, e.g. $\backslash underseta$, which is a convention for indicating boldface type. If the vector represents a directed distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

or German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...

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 Antiqu ...

letters such as $\backslash mathfrak$.
Vectors are usually shown in graphs or other diagrams as arrows (directed line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

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, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

to the plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...

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. 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 fin-like stabilizers c ...

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 vectors in a Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

. The endpoint of a vector can be identified with an ordered list of ''n'' real numbers (''n''-tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

). These numbers are the coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

of the endpoint of the vector, with respect to a given Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, 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
$$\backslash mathbf\; =\; (2,3).$$
The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation $\backslash overrightarrow$ is usually deemed not necessary (and is indeed rarely used).
In ''three dimensional'' Euclidean space (or ), vectors are identified with triples of scalar components:
$$\backslash mathbf\; =\; (a\_1,\; a\_2,\; a\_3).$$
also written,
$$\backslash mathbf\; =\; (a\_x,\; a\_y,\; a\_z).$$
This can be generalised to ''n-dimensional'' Euclidean space (or ).
$$\backslash mathbf\; =\; (a\_1,\; a\_2,\; a\_3,\; \backslash cdots,\; a\_,\; a\_n).$$
These numbers are often arranged into a column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...

or row vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...

, particularly when dealing with matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

, as follows:
$$\backslash mathbf\; =\; \backslash begin\; a\_1\backslash \backslash \; a\_2\backslash \backslash \; a\_3\backslash \backslash \; \backslash end\; =;\; href="/html/ALL/l/a\_1\backslash \_a\_2\backslash \_a\_3\_.html"\; ;"title="a\_1\backslash \; a\_2\backslash \; a\_3\; ">a\_1\backslash \; a\_2\backslash \; a\_3$$
Another way to represent a vector in ''n''-dimensions is to introduce the standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...

vectors. For instance, in three dimensions, there are three of them:
$$\_1\; =\; (1,0,0),\backslash \; \_2\; =\; (0,1,0),\backslash \; \_3\; =\; (0,0,1).$$
These have the intuitive interpretation as vectors of unit length pointing up the ''x''-, ''y''-, and ''z''-axis of a vector component
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...

s (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes ''x'', ''y'', and ''z'' (see figure), while ''a''unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...

s). In this case, the scalar and vector components are denoted respectively ''aindex notation
In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to th ...

and the summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...

commonly used in higher level mathematics, physics, and engineering.
Decomposition or resolution

As explained above, a vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the 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 $\backslash mathbf,\; \backslash mathbf,\; \backslash mathbf$ as abasis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...

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
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...

($\backslash boldsymbol,\; \backslash boldsymbol,\; \backslash mathbf$) or spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

($\backslash mathbf,\; \backslash boldsymbol,\; \backslash boldsymbol$). 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 de ...

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 component
In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the no ...

s'' of a vector relate to the ''radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

'' of an object. The former is parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...

to the radius and the latter is orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

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
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...

).
Basic properties

The following section uses theEquality

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 $$a\_1\; =\; b\_1,\backslash quad\; a\_2=b\_2,\backslash quad\; a\_3=b\_3.\backslash ,$$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 $$a\_1\; =\; -b\_1,\backslash quad\; a\_2=-b\_2,\backslash quad\; a\_3=-b\_3.\backslash ,$$ 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.Addition and subtraction

The sum of a and b of two vectors may be defined as $$\backslash mathbf+\backslash mathbf\; =(a\_1+b\_1)\backslash mathbf\_1\; +(a\_2+b\_2)\backslash mathbf\_2\; +(a\_3+b\_3)\backslash mathbf\_3.$$ The resulting vector is sometimes called the resultant vector of a and b. 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 aScalar multiplication

A vector may also be multiplied, or re-''scaled'', by areal number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

''r''. In the context of 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
$$r\backslash mathbf=(ra\_1)\backslash mathbf\_1\; +(ra\_2)\backslash mathbf\_2\; +(ra\_3)\backslash mathbf\_3.$$
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 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
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

'' or ''norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...

'' of the vector a is denoted by ‖a‖ or, less commonly, , a, , which is not to be confused with the absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

(a scalar "norm").
The length of the vector a can be computed with the ''Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...

'',
$$\backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; =\backslash sqrt,$$
which is a consequence of the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

since the basis vectors eUnit 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: $$\backslash mathbf\; =\; \backslash frac\; =\; \backslash frac\backslash mathbf\_1\; +\; \backslash frac\backslash mathbf\_2\; +\; \backslash frac\backslash mathbf\_3$$Zero vector

The ''zero vector'' is the vector with length zero. Written out in coordinates, the vector is , and it is commonly denoted $\backslash vec$, 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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

'', or, since its result is a scalar, the ''scalar product'') is denoted by a ∙ b, and is defined as:
$$\backslash mathbf\backslash cdot\backslash mathbf\; =\backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; \backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; \backslash cos\backslash theta,$$
where ''θ'' is the measure of the angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...

between a and b (see trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

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
$$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; a\_1\; b\_1\; +\; a\_2\; b\_2\; +\; a\_3\; b\_3.$$
Cross product

The ''cross product'' (also called the ''vector product'' or ''outer product'') is only meaningful in three orseven
7 is a number, numeral, and glyph.
7 or seven may also refer to:
* AD 7, the seventh year of the AD era
* 7 BC, the seventh year before the AD era
* The month of
July
Music Artists
* Seven (Swiss singer) (born 1978), a Swiss recording artist ...

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
$$\backslash mathbf\backslash times\backslash mathbf\; =\backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; \backslash left\backslash ,\; \backslash mathbf\backslash right\backslash ,\; \backslash sin(\backslash theta)\backslash ,\backslash mathbf$$
where ''θ'' is the measure of the angle between a and b, and n is a unit vector perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

to both a and b which completes a right-handed
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...

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
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

). This is the right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...

.
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
$$\backslash times\; =\; (a\_2\; b\_3\; -\; a\_3\; b\_2)\; \_1\; +\; (a\_3\; b\_1\; -\; a\_1\; b\_3)\; \_2\; +\; (a\_1\; b\_2\; -\; a\_2\; b\_1)\; \_3.$$
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 and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...

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: $$(\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf)\; =\backslash mathbf\backslash cdot(\backslash mathbf\backslash times\backslash mathbf).$$ It has three primary uses. First, the absolute value of the box product is the volume of theparallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...

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 linearly dependent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...

, 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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

of the 3-by-3 matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

having the three vectors as rows
$$(\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf)=\backslash begin\; a\_1\; \&\; a\_2\; \&\; a\_3\; \backslash \backslash \; b\_1\; \&\; b\_2\; \&\; b\_3\; \backslash \backslash \; c\_1\; \&\; c\_2\; \&\; c\_3\; \backslash \backslash \; \backslash end$$
The scalar triple product is linear in all three entries and anti-symmetric in the following sense:
$$(\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf)\; =\; (\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf)\; =\; (\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf)=\; -(\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf)\; =\; -(\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf)\; =\; -(\backslash mathbf\backslash \; \backslash mathbf\backslash \; \backslash mathbf).$$
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 $$\backslash mathbf\; =\; p\backslash mathbf\_1\; +\; q\backslash mathbf\_2\; +\; r\backslash mathbf\_3.$$ The scalar components in the ''e'' basis are, by definition, $$\backslash begin\; p\; \&=\; \backslash mathbf\backslash cdot\backslash mathbf\_1,\; \backslash \backslash \; q\; \&=\; \backslash mathbf\backslash cdot\backslash mathbf\_2,\; \backslash \backslash \; r\; \&=\; \backslash mathbf\backslash cdot\backslash mathbf\_3.\; \backslash end$$ In another orthonormal basis ''n'' = that is not necessarily aligned with ''e'', the vector a is expressed as $$\backslash mathbf\; =\; u\backslash mathbf\_1\; +\; v\backslash mathbf\_2\; +\; w\backslash mathbf\_3$$ and the scalar components in the ''n'' basis are, by definition, $$\backslash begin\; u\; \&=\; \backslash mathbf\backslash cdot\backslash mathbf\_1,\; \backslash \backslash \; v\; \&=\; \backslash mathbf\backslash cdot\backslash mathbf\_2,\; \backslash \backslash \; w\; \&=\; \backslash mathbf\backslash cdot\backslash mathbf\_3.\; \backslash end$$ 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 $$\backslash begin\; u\; \&=\; (p\backslash mathbf\_1\; +\; q\backslash mathbf\_2\; +\; r\backslash mathbf\_3)\backslash cdot\backslash mathbf\_1,\; \backslash \backslash \; v\; \&=\; (p\backslash mathbf\_1\; +\; q\backslash mathbf\_2\; +\; r\backslash mathbf\_3)\backslash cdot\backslash mathbf\_2,\; \backslash \backslash \; w\; \&=\; (p\backslash mathbf\_1\; +\; q\backslash mathbf\_2\; +\; r\backslash mathbf\_3)\backslash cdot\backslash mathbf\_3.\; \backslash end$$ Distributing the dot-multiplication gives $$\backslash begin\; u\; \&=\; p\backslash mathbf\_1\backslash cdot\backslash mathbf\_1\; +\; q\backslash mathbf\_2\backslash cdot\backslash mathbf\_1\; +\; r\backslash mathbf\_3\backslash cdot\backslash mathbf\_1,\; \backslash \backslash \; v\; \&=\; p\backslash mathbf\_1\backslash cdot\backslash mathbf\_2\; +\; q\backslash mathbf\_2\backslash cdot\backslash mathbf\_2\; +\; r\backslash mathbf\_3\backslash cdot\backslash mathbf\_2,\; \backslash \backslash \; w\; \&=\; p\backslash mathbf\_1\backslash cdot\backslash mathbf\_3\; +\; q\backslash mathbf\_2\backslash cdot\backslash mathbf\_3\; +\; r\backslash mathbf\_3\backslash cdot\backslash mathbf\_3.\; \backslash end$$ Replacing each dot product with a unique scalar gives $$\backslash begin\; u\; \&=\; c\_p\; +\; c\_q\; +\; c\_r,\; \backslash \backslash \; v\; \&=\; c\_p\; +\; c\_q\; +\; c\_r,\; \backslash \backslash \; w\; \&=\; c\_p\; +\; c\_q\; +\; c\_r,\; \backslash end$$ and these equations can be expressed as the single matrix equation $$\backslash begin\; u\; \backslash \backslash \; v\; \backslash \backslash \; w\; \backslash \backslash \; \backslash end\; =\; \backslash begin\; c\_\; \&\; c\_\; \&\; c\_\; \backslash \backslash \; c\_\; \&\; c\_\; \&\; c\_\; \backslash \backslash \; c\_\; \&\; c\_\; \&\; c\_\; \backslash end\; \backslash begin\; p\; \backslash \backslash \; q\; \backslash \backslash \; r\; \backslash end.$$ 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''transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...

from ''e'' to ''n''", or the "rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\en ...

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 In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\en ...

are such that its inverse is equal to its transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...

. 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
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189 ...

or a quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

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 $$\backslash begin\; (a\_1\_1\; +\; a\_2\_2\; +\; a\_3\_3\; +\; a\_4\_4)\; \&+\; (b\_1\_1\; +\; b\_2\_2\; +\; b\_3\_3\; +\; b\_4\_4)\; =\backslash \backslash \; (a\_1+b\_1)\_1\; +\; (a\_2+b\_2)\_2\; \&+\; (a\_3+b\_3)\_3\; +\; (a\_4+b\_4)\_4.\; \backslash end$$ The cross product does not readily generalise to other dimensions, though the closely relatedexterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...

does, whose result is a bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...

. In two dimensions this is simply a pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...

$$(a\_1\_1\; +\; a\_2\_2)\backslash wedge(b\_1\_1\; +\; b\_2\_2)\; =\; (a\_1\; b\_2\; -\; a\_2\; b\_1)\backslash mathbf\_1\; \backslash mathbf\_2.$$
A seven-dimensional cross product
In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in .
Like the cross product in three dimensions, the seven-dim ...

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 adimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

scale. If it represents, for example, a force, the "scale" is of 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
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...

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 representation of the trajectory of the particle. Vector-valued functions can be differentiated and integrated by differentiating or integrating the components of the vector, and many of the familiar rules fromcalculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

continue to hold for the derivative and integral of vector-valued functions.
Position, velocity and acceleration

The position of a point x = (''x''length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

.
Given two points x = (''x''speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...

. For constant velocity the position at time ''t'' will be
$$\_t=\; t\; +\; \_0,$$
where xtime derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t.
Notation
A variety of notations are used to denote th ...

of position. Its dimensions are length/time.
Acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...

a of a point is vector which is the time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t.
Notation
A variety of notations are used to denote th ...

of velocity. Its dimensions are length/timeForce, energy, work

Force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

is a vector with dimensions of mass×length/timeNewton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...

is the scalar multiplication
$$=\; m$$
Work is the dot product of 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 acoordinate transformation
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

. A ''contravariant vector'' is required to have components that "transform opposite to the basis" under changes of basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...

. 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
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...

''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 vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...

, 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 electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...

, momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

, acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...

.
In the language of differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, 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, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

of contravariant rank one. Alternatively, a contravariant vector is defined to be a tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...

, and the rules for transforming a contravariant vector follow from the chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...

.
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
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 de ...

'' of space. A vector which gains a minus sign when the orientation of space changes is called a ''pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...

'' 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 mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

of two ordinary vectors.
One example of a pseudovector is angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

. Driving in a car, and looking forward, each of the wheel
A wheel is a circular component that is intended to rotate on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machines. Wheels, in conjunction wi ...

s 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 that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

, torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...

, 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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

properties. See parity (physics)
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...

.
See also

*Affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...

, which distinguishes between vectors and points
* Array (data structure)
In computer science, an array is a data structure consisting of a collection of ''elements'' (values or variables), each identified by at least one ''array index'' or ''key''. An array is stored such that the position of each element can be co ...

* Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

* Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...

* Complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

* Coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

* Covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation ...

* Four-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...

, a non-Euclidean vector in Minkowski space (i.e. four-dimensional spacetime), important in relativity
* Function space
* 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 mathematician. He was also a physicist, general scholar, and publi ...

's ''Ausdehnungslehre''
* Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

* Normal vector
* Null vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real bilinear forms, definite quadratic forms an ...

* Position (geometry)
* Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...

* Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

* Tangential and normal components (of a vector)
* Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...

* Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...

* Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...

* Vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...

* Vector notation
In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space.
For representing a vector, the common typographic convention is l ...

* Vector-valued function
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...

Notes

References

Mathematical treatments

* * *. *. *. *. * *Physical treatments

* *External links

*Online vector identities

(

PDF
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...

)
Introducing Vectors

A conceptual introduction (

applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...

)
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Kinematics
Abstract algebra
Vector calculus
Linear algebra
Concepts in physics
Vectors (mathematics and physics)
Analytic geometry
Euclidean geometry