This article relates to the vector as a to be mathematical . For the other significances of the word, to see the page of homonymy Vector (homonymy) ----

In Mathematical, the vector is an object conveying more information than the usual numbers, called Scalaire S, and on which it is possible to carry out operations.

In the beginning, a vector is an object of the Euclidean Géométrie. With two points, Euclide associates their distance. However a couple of points carries a load of larger information. They define also a direction and a direction. The vector synthesizes this information.

The concept of vector can be defined in dimension two (the plan) or three (usual Euclidean space). It spreads with spaces of unspecified size. This concept, become abstracted and introduced by a system of Axiom S, is the base of the branch of mathematics called Linear algebra.

The vector allows, in Physique, to model sizes which cannot be completely defined by a number or a numerical function only. For example, to specify a displacement, a Speed, a force or a Electric field, the Direction and the direction are essential. The vectors are opposed to the scalar sizes described by a simple number, like the Masse, the Température or the Densité.

History

The concept of vector is the fruit of a long story, started here more than two thousand years. Two families of ideas, initially distinct, are at the origin of formalization. One of it is the Géométrie, treating Length S, of Angle S and measurements of Surface S and Volume S. the other corresponds to the Algèbre, which treats Nombre S, Addition or the Multiplication and more generally of units provided with operations. An old problem of algebra comes us for example from the Egyptian and is expressed in the following way:

One must divide 100 bread round loaves between ten men including/understanding a navigator, a foreman and a guard, all three receiving double share. What is necessary to give to each one?

These two families of ideas are developed independently, to end up converging towards the concept of vector.

Origins of the two concepts

The Greek civilization at that time develops the geometry with an unequalled level. One of the florets is the treaty named the Éléments of Euclide , dating from the III E. It contains formalization, very rigorous for the time, of a geometry, still now called Euclidean. One finds there the definitions of a right , a plan or our physical space of dimension three allowing to model volumes. The properties of the Distance S, the angles, measurements of surfaces and volumes are studied. The theorems founders, like those called Thalès or Pythagore, are clarified and shown.

The algebra there is developed and contains little primarily Arithmétique. The integers and rational are studied like some irrational, i.e. the numbers which are not written in the form of a whole fraction of . The numbers are always strictly positive.

The China develops the first algebraic ideas at the origin of the vectors. An old text, dating problablement from I E: the Last nine Chapters on mathematical art devotes its eighth part to it. It is entitled Fang cheng or rectangular and milked Disposition of a problem now called Système of linear equations. This culture does not remain about it there, Qin Jiushao (1202 - 1261) generalizes this study with numbers different from the entireties or rational. It uses the congruences, inaugurating a step consisting in defining vectors on whole of exotic numbers. It can thus solve problems involved in the calendar and planet alignments with a frightening precision. The method used will be known only at the 19th century in occident, under the name of Pivot of Gauss. This result is sufficiently astonishing so that Libbrecht specifies that:

We should not underestimate the revolutionary opening of Qin, indeed, since the Théorème of the Chinese remainders of Sun Zi, one passes without intermediary to a Algorithme more advanced than the method of Gauss itself, and there is not the least indication of a gradual evolution.

The geometrical aspect does not escape the Chinese mathematicians. The final chapter, the Gou gu comprises an equivalent of the theorem of Thalès and Pythagore.

Convergence of the algebra and the geometry

The existence of bond between what one now calls the algebra and the geometry is old. The Babylonian knew already the algebraic property of the diagonal of a square of with dimensions length a , namely that its square is equal to two . They could moreover calculate this value with a remarkable precision. This bond is also known Greeks and of Chinese.

It is however necessary to await the Arab Civilization to observe a significant progress. Their mathematicians knew work of the Greeks, particularly those of Euclide. The notations used let think that they had also access to work of the first Chinese mathematicians. Progress determining consists in associating with the geometrical plan of the coordinated . Omar Khayyam (1048 - 1131) seeks the solutions of a purely algebraic problem: to find the roots of a Polynomial of the third degree. A frame of reference enables him to visualize these roots like the Abscisse S of the intersections of a Parabole and a hyperbole.

The system of the coordinates is taken again in Europe. The will to control the prospect pushes the Italian painters to study mathematics. Filippo Brunelleschi (1377 - 1446) discovers the laws of the prospect, resulting from a central projection. These results are formalized by Leon Battista Alberti (1404 - 1472) . The theorists of the prospect have multiple talents. Thus Piero della Francesca (towards 1412 - 1492) , author of a treaty on the question, is at the same time painter and mathematician. Giorgio Vasari (1511 - 1574) indicates, in connection with its talents of geometrician he was not lower than anybody his time and perhaps from time immemorial. .

Contributions of physics

Physics is the following engine of convergence between geometry and algebra. In 1604, Galileo Galilei (1564 - 1642) establishes the law of gravity. The illustrations of its notes show the use of a reference mark. Optics is the branch which ends in the most outstanding progress. Pierre de Fermat (1601 - 1665) , which knew the writings of Galileo, and Rene Descartes (1596 - 1650) are written letters about the dioptric one (the way in which the light is reflected on a mirror) and with the Réfraction (the deviation of a luminous ray when it changes medium, for example while passing from the air to water). They conclude that a reference mark is a systematic method making it possible to apprehend all the problems of Euclidean geometry. These results are consigned in a treaty of Descartes. He writes in introduction: How the calculation of arithmetic refers to the operations geometry . For Descartes, calculation of arithmetic means roughly what is now called algebra. This approach is particularly fertile for the study of a branch incipient from mathematics: the analytical Geometry. An example is given by the study of the Cycloïde. This curve describes the trajectory of a point of the surface of a wheel moving without friction on a horizontal ground.

Isaac Newton (1643 - 1627) develops the analytical geometry and uses it in Astronomie. This application is the origin of the use of the term Vecteur. In 1704, an English technical dictionary indicates:

A line drawn since a planet, moving around a center or hearth of an ellipse, to this center or this hearth, is called Vecteur by some authors of the New Astronomy, because this line seems to carry planet around the center.

This term appears in French under the feather of Pierre-Simon Laplace (1749 - 1827) in the expression radius vector , still in an astronomical context. It comes from Latin vector and indicates the driver of a carriage. Its origin is older, it comes from the Indo-European *VAG, or *VAGH and means carriage.

Thus, at the 17th century, the geometrical and algebraic context of the vector is present. On the other hand, no formalization is proposed and the term, if it is used, still indicates a scalar size.

Formalizations

The first formalization is the fruit of a work of several mathematicians during first half of the 19th century. Bernard Bolzano (1781 - 1848) publishes an elementary book containing an axiomatic construction of the geometry similar to that of Euclide, founded on points, right-hand sides and plans. It associates the algebraic operations of addition and multiplication. The projective Geometry, heiress of work on the prospect for the painters of the Italian rebirth, led Jean-Victor Poncelet (1788 - 1867) and Michel Chasles (1793 - 1880) to refine ^et work of Bolzano. August Ferdinand Möbius (1790 - 1868) adds its contribution to the building by developing the system of coordinated barycentric. Lastly, formalization still currently taught, starting from the concepts of bipoint and équipollence, is the work of Giusto Bellavitis (1803 - 1880) .

Another way is explored, purely algebraic. William Rowan Hamilton (1805 - 1865) notices that the complex numbers represent an Euclidean plan. It spends ten years of its life to find an equivalent in dimension three, and ends up finding the body of the Quaternion S, of dimension four in 1843. He proposes two new definitions for the words scalar vector and . A vector is for him an element of a subset of quarternions, of dimension three. He writes:

Un vector is thus… a kind of natural triplet (suggested by the geometry): and consequently we will see that the quaternions offers a simple notation symbolic in form trinomiale ( i.x + j.y + k.z ); what brings back the design and the expression of such a vector to the form nearest possible to this obtained with the Cartesian coordinates and rectangulaires.

This second way, which gives for the first time a significance similar to modern formalizations of the concept of vector, is then specified and enriched. It now consists in defining a vector as an element of a vector Space. Its history is developed in the associated article.

Geometrical approach

The Euclidean Géométrie is the geometry of the plan or space based on the Axiome S of Euclide. The concepts of not, of right, Length, are introduced by the means of axioms. The vector is then a geometrical object built starting from the precedents.

An intuitive visualization of a vector corresponds to a displacement of a point, or to use the precise mathematical term, a Translation. Thus a vector has a length, the distance between the starting point and of arrived, a direction if displacement is not null, it is the line containing the starting point and of arrived and a direction, since the departure until the arrival.

Definition

A vector is represented by a segment directed (a “arrow”) having for ends a starting point and a point of arrival. The site in the plan or space does not have importance, two displacements of two distinct points of origin can correspond to the same vector, only count its length, its direction and its direction. It is thus possible to make it slip freely into the plan, parallel to itself.

A formal definition uses as a preliminary the concept of bipoint . It is defined like a couple of points. The order has an importance: the first point is called origin . Two bipoints ( has , B ) and ( C , D ) are known as équipollent S when ABDC is a Parallélogramme. The relation of équipollence constitutes a Relation of equivalence on the bipoints. A class of equivalence contains all the bipoints whose second member is the image of the first point by displacement.

The class of equivalence of a bipoint ( has , B ) is called vector and is noted $\backslash \; scriptstyle\; \backslash \; overrightarrow\; \{AB\}$. The bipoint ( has , B ) is a representative. Reciprocally, any vector admits several bipoints representatives, of which none is privileged. If an origin is chosen, there exists single a bipoint representing a given vector.

Thus two bipoints ( has , B ) and ( C , D ) are équipollents if and only if they represent the same vector and one can then write the equality

$\backslash \; overrightarrow\; \{AB\}\; =\; \backslash \; overrightarrow\; \{CD\}.$

All bipoints made up of the repetition of the same point: ( has , has ), are équipollents between them, they are the representatives of a vector qualified of no one . It is noted

$\backslash \; overrightarrow\; \{AA\}\; =\; \backslash \; overrightarrow\; \{0\}$.

The theories presenting the vectors as a class of equivalence of bipoints in general note them by a surmounted letter of an arrow.

Length and angle

The length of a bipoint (has, B) is defined like the length of the subjacent segment. Two bipoints équipollents have the same length. All the representatives of a vector U thus have the same length, which is called standard vector $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ and is in general noted $\backslash \; scriptstyle\; ||\backslash \; vec\; \{U\}||$ (one uses also sometimes simply the letters indicating the vector without the arrow, for example U or AB ). A Unit vector is a vector of standard 1. The null vector is of null standard, $\backslash \; scriptstyle\; ||\backslash \; vec\; \{0\}||\; =\; 0$.

The Angle that form two vectors $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ and $\backslash \; scriptstyle\; \backslash \; vec\; \{v\}$ is noted $\backslash \; scriptstyle\; (\backslash \; widehat\; \{\backslash \; vec\; \{U\},\; \backslash \; vec\; \{v\}\})$. It is defined as the angle which two of the same representatives form origin. Thus if ( has , B ) is a representative of $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ and ( has , C ) a representative of $\backslash \; scriptstyle\; \backslash \; vec\; \{v\}$, then

$(\backslash \; widehat\; \{\backslash \; vec\; \{U\},\; \backslash \; vec\; \{v\}\})\; =\; \backslash \; widehat\; \{VAT\}$

In the plan directed, it is possible to define the concept of directed angle of two vectors. It is not the case in space.

Operations

See also: vector Calculus in Euclidean geometry

Geometrical constructions allow the definition of the addition and the multiplication by a scalar. The name given to the operations is the consequence of the similarity with the operations on the numbers (Commutativité, Associativité, Distributivité, presence of a neutral element and absorbent). For this reason, not only the names of the operations but the notations are similar.

If $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ and $\backslash \; scriptstyle\; \backslash \; vec\; \{v\}$ is two vectors, that is to say a couple ( has , B ) of points representing $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ and C the point such that the couple ( B , C ) represents the vector $\backslash \; scriptstyle\; \backslash \; vec\; \{v\}$. Then a representative of the vector $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}\; +\; \backslash \; vec\; \{v\}$ is the couple ( has , C ). If $\backslash \; scriptstyle\; \backslash \; vec\; \{v\}$ is the null vector, then the points B and C are confused, the sum is then equal to $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ and the null vector is well the neutral element for the addition of the vectors. Either α a number, if $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ is the null vector, then α.$\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ is also the null vector, if not there exists a single line containing has and B , and a single point D such as the distance between has and C or equalizes with $\backslash \; scriptstyle\; |\backslash \; alpha|\backslash ;\; .\; ||\backslash \; vec\; \{U\}||$ and it direction of ( has , B ) if α is positive and the reverse if not.

Once equipped with a structure of vector space, the demonstrations of the Euclidean geometry proves often simplified. An example is given by the Théorème from Thalès.

Critical of geometrical formalization

This formalization is most intuitive and simplest. For this reason, it is used to present the vectors in the Secondary education in France.

Nevertheless, such an approach is rigorous only if the plan or space, qualified refines, is defined in a perfect way. The construction of Euclide, if it is didactic, is compartmental. It misses axioms for a complete characterization. It is possible to circumvent this difficulty while using, for example the construction of Hilbert to define the plan or space. It is based on twenty rich and subtle axioms, in the place of the five defined by Euclide. A rigorous approach consequently puts at evil the simplicity of an initial presentation.

It is not the single weakness of geometrical formalization. The development of mathematics widened considerably the fields of application of the vectors. Dimension two of the plan or three of space is not any more one systematically respected constraint. It is not rare to use spaces of unspecified size, sometimes infinite. Generalization with higher dimensions is not single enrichment. The scalars are not always real. There exist certain cases where the complex or even a together finished numbers are adapted. A construction by the bipoints is not simplest to define a vector in all its general information.

Another construction fills this gap. It is founded on two units: one containing the scalars, the other vectors. Second vector Space is called. These two units are provided with operations and of the axioms are checked for each operation. In the case of the plan or of Euclidean space, the two approaches are equivalent. On the other hand, algebraic construction spreads more easily for dimensions and of the unspecified scalars. In against part, it is less intuitive and proceeds of a more complex step. She is taught in the higher in France. This different construction to formalize the same concept of vector is treated in the article entitled vector Space.

Algebraic approach

Coordinates and vectors columns

See also: Base (linear algebra)

In a plan, two vectors $\backslash \; scriptstyle\; \backslash \; vec\; \{has\}$ and $\backslash \; scriptstyle\; \backslash \; vec\; \{B\}$ nonnull and of different directions has an important property. A vector $\backslash \; unspecified\; scriptstyle\; \backslash \; vec\; \{U\}$ is nap of a multiple of $\backslash \; scriptstyle\; \backslash \; vec\; \{has\}$ and $\backslash \; scriptstyle\; \backslash \; vec\; \{B\}$. That means that there exist two single numbers U ₁ and U ₂ such as:

$\backslash \; vec\; \{U\}\; =\; u\_1.\; \backslash \; vec\; \{has\}\; +\; u\_2.\; \backslash \; vec\; \{B\}\; \backslash ;$

$\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ is then described as linear Combinaison of $\backslash \; scriptstyle\; \backslash \; vec\; \{has\}$ and $\backslash \; scriptstyle\; \backslash \; vec\; \{B\}$. As any vector of the plan is expressed in a single way as linear combination of $\backslash \; scriptstyle\; \backslash \; vec\; \{has\}$ and $\backslash \; scriptstyle\; \backslash \; vec\; \{B\}$, the family ($\backslash \; scriptstyle\; \backslash \; vec\; \{has\}$, $\backslash \; scriptstyle\; \backslash \; vec\; \{B\}$) is qualified basic plan and U ₁, U ₂ is called Coordonnée S of the vector $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ in this base. This definition corresponds to that of a plan refines provided with a reference mark.

Such a property is still true in space. However, two vectors are not enough any more, any base contains exactly three nonnull vectors and whose directions are not Coplanaire S (i.e. which there does not exist any plan containing the three directions). So in space, the three coordinates of a vector $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ are U ₁, U ₂ and U ₃, it is of use to note:

$\backslash \; vec\; \{U\}\; \backslash \; begin\; \{pmatrix\}\; u\_1\; \backslash \; \backslash \; u\_2\; \backslash \; \backslash \; u\_3\; \backslash \; end\; \{pmatrix\}$

to indicate the coordinates of the vector. The table is called vector-column and corresponds to a particular case of matrix. The algebraic operations on the vectors are simple, with such a representation. To add two vectors amounts adding each coordinate and the multiplication by a scalar amounts multiplying each coordinates by the scalar.

In a vectorial plan, a vector is identified with a couple of scalars, and in space with a triplet. If the selected numbers are real then a plan (respectively a space) identifies with R ² (respectively with R ³). Here, R indicates the whole of the real numbers.

Outline of an algebraic construction

See also: vector Space

The preceding logic, applied for a dimension equalizes to two or three spreads. It is thus possible to consider the structure R ⁿ where in a more general way K ⁿ with K a whole of scalars having of good properties (precisely, K is a body Commutatif). Such a structure has an addition, and a scalar multiplication definite as in the preceding paragraph.

It is possible to still generalize the definition of a vector. If a unit E has an addition and a scalar multiplication on a commutative body and if its operations check certain properties, called axioms and described in the detailed article, then E is called vector space and an element of E vector.

Very many examples of mathematically interesting units have such a structure. It is the case for example spaces of Polynôme S, functions checking certain properties of regularity, of matrices… All these units can then be studied with the tools of the vector calculus and the Linear algebra.

The concept of dimension provides the first result of classification concerning the vector spaces. In a vector space of finished size N , it is possible, with the help of the choice of a base, to be reduced to calculation on vectors columns of size N . There exist also vector spaces of infinite size. The whole of the functions of R in R is thus a vector space on the body of the real numbers, of infinite size. Seen under this angle, such a function is a vector.

Algebraic construction and geometry

If two constructions, algebraic and geometrical are equivalent for the vectorial structures of the plan and of usual space, the geometry brings in more the concepts of distance and angle.

The scalar concept of Produit makes it possible to fill this gap. A scalar product associates with two vectors a reality, if the two vectors are identical, then reality is positive. There exists a scalar product such that the standard of the vector is equal to the square Racine scalar product of the vector and itself. The Euclidean geometry then seems the study of a Espace refines associated with a vector space with dimension two or three on the body with realities, provided with a scalar product: plan refines Euclidean or spaces refines Euclidean.

Once equipped with a scalar product, it becomes possible to define on the vector space of the traditional transformations of Euclidean geometry like the symmetry, the rotation or the orthogonal Projection. The transformation associated with the vector spaces always leaves invariant the null vector. Rotations make it possible to define the concept of angle for the vectors. The angle $\backslash \; scriptstyle\; (\backslash \; widehat\; \{\backslash \; vec\; \{U\},\; \backslash \; vec\; \{v\}\})$ is equal to $\backslash \; scriptstyle\; (\backslash \; widehat\; \{\backslash \; vec\; \{u\text{'}\},\; \backslash \; vec\; \{v\text{'}\}\})$ if and only if there exists a rotation which sends $\backslash \; scriptstyle\; \backslash \; vec\; \{U\}$ on $\backslash \; scriptstyle\; \backslash \; vec\; \{u\text{'}\}$ and $\backslash \; scriptstyle\; \backslash \; vec\; \{v\}$ on $\backslash \; scriptstyle\; \backslash \; vec\; \{v\text{'}\}$. This definition, which applies to an algebraic formalization of the concept of vector space, is equivalent to that of geometrical construction. Such an approach sometimes largely simplifies the demonstrations, an example is the Théorème of Pythagore.

The algebraic approach makes it possible to define all the notions of the Euclidean geometry, it generalizes this geometry with an unspecified dimension if the numbers are real. In the case of the complex numbers a similar construction, called square Space, exists.

Uses of the vectors

The examples quoted in this article are relatively simple and didactic. Other cases, more general are presented in the articles vector Space and Linear algebra.

Mathematics

A vast part of mathematics use the vectors, in algebra in geometry or analysis.

An archetypal example in algebra is the resolution of a Système of linear equations. An example of three equations to three unknown factors corresponds to the research of the vectors of dimension three, Antécédent S of a Linear application of a given vector. The Euclidean plan can also be represented by the Plan complex. The canonical base is made up of two vectors the unit of realities and the pure Imaginary number.

The vectors offer an effective tool for the resolution of many problems of geometry. They are used for the determination of properties of parallelism or Orthogonalité of right-hand sides, plan or segments. Through the use of the barycentric coordinated , the vectors form a tool adapted to characterize the center of a geometrical figure and allow a simple demonstration of the Théorème of Leibniz, Théorème of Ceva like many results on the geometry of the triangles. The scalar product, which is particularly simply expressed in a orthonormée Base, offers many possibilities. It allows, for example, to measure the distance from a point on a line or a plan. Such a base makes it possible to express also simply geometrical transformations like the orthogonal Projection on a plan or a line.

The analysis is not in remainder. The vector space R ², copy of the Euclidean plan is the natural framework of representation of the Graphe of a function. The vectors make it possible for example to determine the line perpendicular to a curve in order to determine the hearths of a Conique. The chart offers a solution to determine an approximation of a root of a equation if a resolution by an algebraic method is not known.

Physics

See also: Mechanical of the point, Force (physical)

The Physique is at the origin of the term of vector, it always largely uses this concept. The historical reason comes owing to the fact that in traditional Physique the space which surrounds us is well modelled like Espace refines (Euclidean geometry) of dimension three with time (absolute) like parameter of evolution. In physics, an addition of vectors can have direction only if their respective coordinates have same the dimension.

The position of a point is described by coordinates in a reference mark, but its speed and its acceleration are vectors. To establish the Mechanical point, i.e. the motion study of a material point, the vectors are essential. The position of a point is modelled by its three coordinates (which are real numbers) whose each one is a function of time; one can also describe it by the vector position energy of the origin of the reference mark at the point: the coordinates of the vector are then the same ones as those of the point. The vector Speed is equal to derived from the vector position (i.e.: the coordinates of the Flight Path Vector are the derivative of those of the vector position), and it is still a vector. It is the same for the Accélération, corresponding to the derived second.

In a Reference mark galiléen, the acceleration of a point is proportional to the force which is applied to him. A force is also a vector. The trajectory of a planet is known by the force which is applied to him at every moment. This force is the consequence of the Gravitation, primarily due to the sun. This phenomenon is described by the data of the gravitational Champ. This field associates a vector proportional to the force of gravitation with each point of space.

This modeling puts up with difficulty with the restricted Relativité owing to the fact that the changes of reference frames do not depend linearly speed there, and it does not relate to the General relativity which does not use Euclidean space (except for approximations). In Quantum physics the coordinates can be that of a particle only by taking account of the Principe of uncertainty, and the forces are due to exchanges of particles. electricity and the Magnétisme act like the Gravitation. The magnetic fields or electric allow, using the resolution of a differential equation, to determine the trajectory of a point. -->

Generalizations

Mathematics

The linear applications are functions of a vector space in another and respecting the addition and the external multiplication. They are added and multiplied scalairement, and thus have the properties which make they vectors. It is the same for the matrices. A matrix always has the properties of a vector, even if it is not of column type.

The two preceding examples correspond to cases where the structure is enriched by an internal multiplication. It bears the name of algebra, its elements are often called vectors and sometimes points. Examples are given by the whole of the polynomials to real coefficients or a Algèbre of Dregs.

In other cases, the structure is impoverished. A module is a similar structure such as the scalars different from zero are not always invertible. The term of vector nevertheless is always used.

Physics

thumb|200 px|According to the point of application of the forces, the solid rocks or not. The associated mathematical object is a slipping vector.

The laws establishing the movements of a point applies a solid also in the case of, calculations become nevertheless more complex. If the vectors remain omnipresent, the point of application of the force has its importance. According to its position, the solid turns in addition to the displacement of sound Center of gravity. To take account of this phenomenon, of new definitions are proposed. A dependant vector or pointer is a couple made up of a vector and a point called not of application. The rotation of the solid is the consequence of a physical size called moment. It does not depend on the position of the vector on a given line. For this reason, a vector slipping is a couple made up of a vector and a line closely connected. In this context, and to avoid any ambiguity, a vector in the classical sense of the term is called free vector .

To take account at the same time of the rotation and the movement of the center of gravity, a more complex mathematical being is used. It bears the name of Torseur . It corresponds to a vector of dimension six, three coordinates describe the displacement of the center of gravity and the three others the rotation of the solid. The torques have in more one specific law of composition. Physics uses other generalizations, one can quote the Tenseur or the Pseudovecteur.

Data processing

See also: vectorial Image, Table (structure of data)

Data processing uses the term of vector, at the same time for geometrical ratios and algebraic. The representation of an image on a screen of computer uses with the choice two techniques: matric and vectorial. The first uses graphic elements definite points by point. With each Pixel the quantity with primary colors is associated associated. If this method is economic in term of and computing power memory, an enlarging of the size of the image has for consequence a effect of staircase.

A vectorial drawing is a representation made up of geometrical objects (lines, points, polygons, curves,…) having attributes of form, position, color, etc To the difference of the preceding technique, against a more expensive method in term of memory and computing power, there does not exist effect of staircase.

Representation of data in data processing, for functions of memory or of calculation, bases on tables of Byte S. If a byte is identified with a scalar, which this conceives because two bytes are added and multiplied, then such a table are connected with a family of coordinates. For this reason, such a table is called vector. By extension, the term of vector indicates also tables whose coordinates are other thing that numbers, for example of the pointer or unspecified data-processing structures.

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