A vector is what is needed to "carry" the pointA to the pointB; the Latin wordvector means 'carrier'.[4] It was first used by 18th centuryastronomers investigating planetary revolution around the Sun.[5] The magnitude of the vector is the distance between the two points, and the direction refers to the direction ofdisplacement fromA toB. Manyalgebraic operations onreal numbers such asaddition,subtraction,multiplication, andnegation have close analogues for vectors,[6] operations which obey the familiar algebraic laws ofcommutativity,associativity, anddistributivity. These operations and associated laws qualifyEuclidean vectors as an example of the more generalized concept of vectors defined simply as elements of avector space.
Vectors play an important role inphysics: thevelocity andacceleration of a moving object and theforces acting on it can all be described with vectors.[7] Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example,position ordisplacement), 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 thecoordinate system used to describe it. Other vector-like objects that describephysical quantities and transform in a similar way under changes of the coordinate system includepseudovectors andtensors.[8]
The vector concept, as it is known 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.[9] In 1835,Giusto Bellavitis abstracted the basic idea when he established the concept ofequipollence. Working in a Euclidean plane, he made equipollent any pair ofparallel line segments of the same length and orientation. Essentially, he realized anequivalence relation on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane.[9]: 52–4 The termvector was introduced byWilliam Rowan Hamilton as part of aquaternion, which is a sumq =s +v of areal numbers (also calledscalar) and a 3-dimensionalvector. Like Bellavitis, Hamilton viewed vectors as representative ofclasses of equipollent directed segments. Ascomplex numbers use animaginary unit to complement thereal line, Hamilton considered the vectorv to be theimaginary part of a quaternion:[10]
The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.
Several other mathematicians developed vector-like systems in the middle of the nineteenth century, includingAugustin Cauchy,Hermann Grassmann,August Möbius,Comte de Saint-Venant, andMatthew O'Brien. Grassmann's 1840 workTheorie 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.[9]Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867Elementary Treatise of Quaternions included extensive treatment of the nabla ordel operator ∇. In 1878,Elements of Dynamic was published byWilliam Kingdon Clifford. Clifford simplified the quaternion study by isolating thedot product andcross product 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, who was exposed to quaternions throughJames Clerk Maxwell'sTreatise on Electricity and Magnetism, separated off their vector part for independent treatment. The first half of Gibbs'sElements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis.[9][6] In 1901,Edwin Bidwell Wilson publishedVector Analysis, adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus.
Inphysics andengineering, a vector is typically regarded as a geometric entity characterized by amagnitude and arelative direction. It is formally defined as adirected line segment, or arrow, in aEuclidean space.[11] Inpure mathematics, avector is defined more generally as any element of avector space. 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 abstract vectors, as they are elements of a special kind of vector space calledEuclidean space. 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 asgeometric,spatial, orEuclidean vectors.
A Euclidean vector may possess a definiteinitial point andterminal point; such a condition may be emphasized calling the result abound vector.[12] When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called afree vector. The distinction between bound and free vectors is especially relevant in mechanics, where aforce applied to a body has a point of contact (seeresultant force andcouple).
Two arrows and in space represent the same free vector if they have the same magnitude and direction: that is, they areequipollent if the quadrilateralABB′A′ is aparallelogram. If the Euclidean space is equipped with a choice oforigin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.
The termvector also has generalizations to higher dimensions, and to more formal approaches with much wider applications.
In classicalEuclidean geometry (i.e.,synthetic geometry), vectors were introduced (during the 19th century) asequivalence classes underequipollence, ofordered pairs of points; two pairs(A,B) and(C,D) being equipollent if the pointsA,B,D,C, in this order, form aparallelogram. Such an equivalence class is called avector, more precisely, a Euclidean vector.[13] The equivalence class of(A,B) is often denoted
A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of theline segment(A,B)) and same direction (e.g., the direction fromA toB).[14] 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 toscalars, which have no direction.[7] For example,velocity,forces andacceleration are represented by vectors.
In modern geometry, Euclidean spaces are often defined fromlinear algebra. More precisely, a Euclidean spaceE is defined as a set to which is associated aninner product space of finite dimension over the reals and agroup action of theadditive group of which isfree andtransitive (SeeAffine space for details of this construction). The elements of are calledtranslations. 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 thereal coordinate space equipped with thedot product. This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product.
The Euclidean space is often presented asthestandard Euclidean space of dimensionn. This is motivated by the fact that every Euclidean space of dimensionn isisomorphic to the Euclidean space More precisely, given such a Euclidean space, one may choose any pointO as anorigin. ByGram–Schmidt process, one may also find anorthonormal basis 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 definesCartesian coordinates of any pointP of the space, as the coordinates on this basis of the vector These choices define an isomorphism of the given Euclidean space onto by mapping any point to then-tuple of its Cartesian coordinates, and every vector to itscoordinate vector.
Since the physicist's concept offorce has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward forceF of 15newtons. If the positiveaxis is also directed rightward, thenF is represented by the vector 15 N, and if positive points leftward, then the vector forF is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δs of 4meters would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless.
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, the magnitude of which isspeed. For instance, the velocity5 meters per second upward could be represented by the vector (0, 5) (in 2 dimensions with the positivey-axis as 'up'). Another quantity represented by a vector isforce, since it has a magnitude and direction and follows the rules of vector addition.[7] Vectors also describe many other physical quantities, such as linear displacement,displacement, linear acceleration,angular acceleration,linear momentum, andangular momentum. Other physical vectors, such as theelectric andmagnetic field, are represented as a system of vectors at each point of a physical space; that is, avector 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 theCartesian coordinate system, a bound vector can be represented by identifying the coordinates of its initial and terminal point. For instance, the pointsA = (1, 0, 0) andB = (0, 1, 0) in space determine the bound vector pointing from the pointx = 1 on thex-axis to the pointy = 1 on they-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 originO = (0, 0, 0). 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 positivex-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
In the geometrical and physical settings, it is sometimes possible to associate, in a natural way, alength or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of anangle between two vectors. If thedot product 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 thecross product, which supplies an algebraic characterization of thearea andorientation in space of theparallelogram defined by two vectors (used as sides of the parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define theexterior product, which (among other things) supplies an algebraic characterization of the area and orientation in space of then-dimensionalparallelotope defined byn vectors.
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 ofvector spaces (for free vectors) andaffine spaces (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes fromthermodynamics, where many quantities of interest can be considered vectors in a space with no notion of length or angle.[15]
In physics, as well as mathematics, a vector is often identified with atuple of components, or list of numbers, that act as scalar coefficients for a set ofbasis vectors. 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 calledcovariant orcontravariant, 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 asgradient. If you change units (a special case of achange 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, seecovariance and contravariance of vectors).Tensors are another type of quantity that behave in this way; a vector is one type oftensor.
In puremathematics, a vector is any element of avector space over somefield and is often represented as acoordinate 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".
Vectors are usually denoted inlowercase boldface, as in, and, or in lowercase italic boldface, as ina. (Uppercase letters are typically used to representmatrices.) Other conventions include ora, especially in handwriting. Alternatively, some use atilde (~) or a wavy underline drawn beneath the symbol, e.g., which is a convention for indicating boldface type. If the vector represents a directeddistance ordisplacement from a pointA to a pointB (see figure), it can also be denoted as orAB. InGerman literature, it was especially common to represent vectors with smallfraktur letters such as.
Vectors are usually shown in graphs or other diagrams as arrows (directedline segments), as illustrated in the figure. Here, the pointA is called theorigin,tail,base, orinitial point, and the pointB is called thehead,tip,endpoint,terminal point orfinal point. The length of the arrow is proportional to the vector'smagnitude, while the direction in which the arrow points indicates the vector's direction.
On a two-dimensional diagram, a vectorperpendicular to theplane 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 anarrow head on and viewing the flights of an arrow from the back.
A vector in the Cartesian plane, showing the position of a pointA with coordinates (2, 3).
In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in ann-dimensional Euclidean space can be represented ascoordinate vectors in aCartesian coordinate system. The endpoint of a vector can be identified with an ordered list ofn real numbers (n-tuple). These numbers are thecoordinates of the endpoint of the vector, with respect to a givenCartesian coordinate system, and are typically called thescalar components (orscalar projections) of the vector on the axes of the coordinate system.
As an example in two dimensions (see figure), the vector from the originO = (0, 0) to the pointA = (2, 3) is simply written as
The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation is usually deemed not necessary (and is indeed rarely used).
Inthree dimensional Euclidean space (orR3), vectors are identified with triples of scalar components:also written,
This can be generalised ton-dimensional Euclidean space (orRn).
Another way to represent a vector inn-dimensions is to introduce thestandard basis vectors. For instance, in three dimensions, there are three of them:These have the intuitive interpretation as vectors of unit length pointing up thex-,y-, andz-axis of aCartesian coordinate system, respectively. In terms of these, any vectora inR3 can be expressed in the form:
or
wherea1,a2,a3 are called thevector components (orvector projections) ofa on the basis vectors or, equivalently, on the corresponding Cartesian axesx,y, andz (see figure), whilea1,a2,a3 are the respectivescalar components (or scalar projections).
In introductory physics textbooks, the standard basis vectors are often denoted instead (or, in which thehat symbol typically denotesunit vectors). In this case, the scalar and vector components are denoted respectivelyax,ay,az, andax,ay,az (note the difference in boldface). Thus,
The notationei is compatible with theindex notation and thesummation convention commonly used in higher level mathematics, physics, and engineering.
As explainedabove, a vector is often described by a set of vector components thatadd up to form the given vector. Typically, these components are theprojections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to bedecomposed orresolved with respect to that set.
Illustration of tangential and normal components of a vector to a surface.
The decomposition or resolution[16] 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 as abasis 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 acylindrical coordinate system () orspherical coordinate system (). 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 theirorientation as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectivelynormal, andtangent to a surface (see figure). Moreover, theradial andtangential components of a vector relate to theradius ofrotation of an object. The former isparallel to the radius and the latter isorthogonal to it.[17]
In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., aglobal coordinate system, orinertial reference frame).
The following section uses theCartesian coordinate system with basis vectorsand assumes that all vectors have the origin as a common base point. A vectora will be written as
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 vectorsandare equal if
Two vectors areopposite if they have the same magnitude butopposite direction;[18] so two vectors
and
are opposite if
Two vectors areequidirectional (orcodirectional) if they have the same direction but not necessarily the same magnitude.[18]Two vectors areparallel if they have either the same or opposite direction, but not necessarily the same magnitude; two vectors areantiparallel if they have strictly opposite direction, but not necessarily the same magnitude.[a]
The sum ofa andb of two vectors may be defined asThe resulting vector is sometimes called theresultant vector ofa andb.
The addition may be represented graphically by placing the tail of the arrowb at the head of the arrowa, and then drawing an arrow from the tail ofa to the head ofb. The new arrow drawn represents the vectora +b, as illustrated below:[7]
The addition of two vectorsa andb
This addition method is sometimes called theparallelogram rule becausea andb form the sides of aparallelogram anda +b is one of the diagonals. Ifa andb are bound vectors that have the same base point, this point will also be the base point ofa +b. One can check geometrically thata +b =b +a and (a +b) +c =a + (b +c).
The difference ofa andb is
Subtraction of two vectors can be geometrically illustrated as follows: to subtractb froma, place the tails ofa andb at the same point, and then draw an arrow from the head ofb to the head ofa. This new arrow represents the vector(-b) +a, with(-b) being the opposite ofb, see drawing. And(-b) +a =a −b.
Scalar multiplication of a vector by a factor of 3 stretches the vector out.
A vector may also be multiplied, or re-scaled, by anyreal numberr. In the context ofconventional vector algebra, these real numbers are often calledscalars (fromscale) to distinguish them from vectors. The operation of multiplying a vector by a scalar is calledscalar multiplication. The resulting vector is
Intuitively, multiplying by a scalarr stretches a vector out by a factor ofr. Geometrically, this can be visualized (at least in the case whenr is an integer) as placingr copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.
Ifr is negative, then the vector changes direction: it flips around by an angle of 180°. Two examples (r = −1 andr = 2) are given below:
The scalar multiplications −a and 2a of a vectora
Scalar multiplication isdistributive over vector addition in the following sense:r(a +b) =ra +rb for all vectorsa andb and all scalarsr. One can also show thata −b =a + (−1)b.
Thelength,magnitude ornorm of the vectora is denoted by ‖a‖ or, less commonly, |a|, which is not to be confused with theabsolute value (a scalar "norm").
The length of the vectora can be computed with theEuclidean norm,
which is a consequence of thePythagorean theorem since the basis vectorse1,e2,e3 are orthogonal unit vectors.
This happens to be equal to the square root of thedot product, discussed below, of the vector with itself:
Aunit 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.[14] This is known asnormalizing a vector. A unit vector is often indicated with a hat as inâ.
To normalize a vectora = (a1,a2,a3), scale the vector by the reciprocal of its length ‖a‖. That is:
Thezero vector is the vector with length zero. Written out in coordinates, the vector is(0, 0, 0), and it is commonly denoted,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 vectora isa (that is,0 +a =a).
Thedot product of two vectorsa andb (sometimes called theinner product, or, since its result is a scalar, thescalar product) is denoted bya ∙ b, and is defined as:
whereθ is the measure of theangle betweena andb (seetrigonometric function for an explanation of cosine). Geometrically, this means thata andb are drawn with a common start point, and then the length ofa is multiplied with the length of the component ofb that points in the same direction asa.
The dot product can also be defined as the sum of the products of the components of each vector as
Thecross product (also called thevector product orouter product) is only meaningful in three orseven 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, denoteda × b, is a vector perpendicular to botha andb and is defined as
whereθ is the measure of the angle betweena andb, andn is a unit vectorperpendicular to botha andb which completes aright-handed system. The right-handedness constraint is necessary because there existtwo unit vectors that are perpendicular to botha andb, namely,n and (−n).
An illustration of the cross product
The cross producta × b is defined so thata,b, anda × b also becomes a right-handed system (althougha andb are not necessarilyorthogonal). This is theright-hand rule.
The length ofa × b can be interpreted as the area of the parallelogram havinga andb as sides.
The cross product can be written as
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 apseudovector instead of a vector (see below).
Thescalar triple product (also called thebox product ormixed 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 (abc) and defined as:
It has three primary uses. First, the absolute value of the box product is the volume of theparallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors arelinearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectorsa,b andc 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 thedeterminant of the 3-by-3matrix having the three vectors as rows
The scalar triple product is linear in all three entries and anti-symmetric in the following sense:
All examples thus far have dealt with vectors expressed in terms of the same basis, namely, thee basis {e1,e2,e3}. 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 thee basis, a vectora is expressed, by definition, as
The scalar components in thee basis are, by definition,
In another orthonormal basisn = {n1,n2,n3} that is not necessarily aligned withe, the vectora is expressed as
and the scalar components in then basis are, by definition,
The values ofp,q,r, andu,v,w relate to the unit vectors in such a way that the resulting vector sum is exactly the same physical vectora 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 expressu,v,w in terms ofp,q,r is to use column matrices along with adirection cosine matrix containing the information that relates the two bases. Such an expression can be formed by substitution of the above equations to form
Distributing the dot-multiplication gives
Replacing each dot product with a unique scalar gives
and these equations can be expressed as the single matrix equation
This matrix equation relates the scalar components ofa in then basis (u,v, andw) with those in thee basis (p,q, andr). Each matrix elementcjk is thedirection cosine relatingnj toek.[19] The termdirection cosine refers to thecosine of the angle between two unit vectors, which is also equal to theirdot product.[19] Therefore,
By referring collectively toe1,e2,e3 as thee basis and ton1,n2,n3 as then basis, the matrix containing all thecjk is known as the "transformation matrix frome ton", or the "rotation matrix frome ton" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "direction cosine matrix frome ton"[19] (because it contains direction cosines). The properties of a rotation matrix are such that itsinverse is equal to itstranspose. This means that the "rotation matrix frome ton" is the transpose of "rotation matrix fromn toe".
The properties of a direction cosine matrix, C are:[20]
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 usingEuler angles or aquaternion 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.[19]
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 asand in four dimensions as
The cross product does not readily generalise to other dimensions, though the closely relatedexterior product does, whose result is abivector. In two dimensions this is simply apseudoscalar
Aseven-dimensional cross product is similar to the cross product in that its result is a vector orthogonal to the two arguments; there is however no natural way of selecting one of the possible such products.
In abstract vector spaces, the length of the arrow depends on adimensionlessscale. If it represents, for example, a force, the "scale" is ofphysical 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 someproportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance.
Often in areas of physics and mathematics, a vector evolves in time, meaning that it depends on a time parametert. For instance, ifr represents the position vector of a particle, thenr(t) gives aparametric representation of the trajectory of the particle. Vector-valued functions can bedifferentiated andintegrated by differentiating or integrating the components of the vector, and many of the familiar rules fromcalculus continue to hold for the derivative and integral of vector-valued functions.
The position of a pointx = (x1,x2,x3) in three-dimensional space can be represented as aposition vector whose base point is the originThe position vector has dimensions oflength.
Given two pointsx = (x1,x2,x3),y = (y1,y2,y3) theirdisplacement is a vectorwhich specifies the position ofy relative tox. The length of this vector gives the straight-line distance fromx toy. Displacement has the dimensions of length.
Thevelocityv of a point or particle is a vector, its length gives thespeed. For constant velocity the position at timet will bewherex0 is the position at timet = 0. Velocity is thetime derivative of position. Its dimensions are length/time.
Accelerationa of a point is vector which is thetime derivative of velocity. Its dimensions are length/time2.
An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under acoordinate transformation. Acontravariant vector is required to have components that "transform opposite to the basis" under changes ofbasis. 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 aninvertible matrixM, so that a coordinate vectorx is transformed tox′ =Mx, then a contravariant vectorv must be similarly transformed viav′ =Mv. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, ifv consists of thex,y, andz-components ofvelocity, thenv 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 abstractvector, 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 includedisplacement,velocity,electric field,momentum,force, andacceleration.
In the language ofdifferential geometry, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining acontravariant vector to be atensor ofcontravariant rank one. Alternatively, a contravariant vector is defined to be atangent vector, and the rules for transforming a contravariant vector follow from thechain rule.
Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flipand gain a minus sign. A transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change theorientation of space. A vector which gains a minus sign when the orientation of space changes is called apseudovector or anaxial vector. Ordinary vectors are sometimes calledtrue vectors orpolar vectors to distinguish them from pseudovectors. Pseudovectors occur most frequently as thecross product of two ordinary vectors.
One example of a pseudovector isangular velocity. Driving in acar, and looking forward, each of thewheels 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, thereflection of this angular velocity vector points to the right, but theactual angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors includemagnetic field,torque, or more generally any cross product of two (true) vectors.
This distinction between vectors and pseudovectors is often ignored, but it becomes important in studyingsymmetry properties.
^Gibbs, J.W. (1901).Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs, by E.B. Wilson, Chares Scribner's Sons, New York, p. 15: "Any vectorr coplanar with two non-collinear vectorsa andb may be resolved into two components parallel toa andb respectively. This resolution may be accomplished by constructing the parallelogram ..."
^Rogers, Robert M. (2007).Applied mathematics in integrated navigation systems (3rd ed.). Reston, Va.: American Institute of Aeronautics and Astronautics.ISBN9781563479274.OCLC652389481.
