Euclidean geometry and Euclidean vector

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. HP G62-115SE Battery Although many of Euclid's results had been stated by earlier mathematicians,[1] Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.[2] The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. HP G62-115SO BatteryIt goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, couched in geometrical language.[3] For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, HP G62-117SO Battery sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Einstein's theory of general relativity is that Euclidean space is a good approximation to the properties of physical space only where the gravitational field is not too strong.[4] HP G62-118EO Battery The Elements Main article: Euclid's Elements The Elements are mainly a systematization of earlier knowledge of geometry. Its superiority over earlier treatments was rapidly recognized, HP G62-120EC Battery with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, e.g., If a triangle has two equal angles, then the sides subtended by the angles are equal. The Pythagorean theorem is provedHP G62-120EE Battery.[5] Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as prime numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved. HP G62-120EG Battery Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. Axioms Euclidean geometry is an axiomatic system, HP G62-120EH Batteryin which all theorems ("true statements") are derived from a small number of axioms.[6] Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):[7] "Let the following be postulated":HP G62-120EK Battery "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line." "To describe a circle with any centre and distance [radius]." "That all right angles are equal to one another." HP G62-120EL Battery The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." HP G62-120EP Battery Although Euclid's statement of the postulates only explicitly asserts the existence of the constructions, they are also taken to be unique. The Elements also include the following five "common notions": Things that are equal to the same thing are also equal to one another. HP G62-120EQ Battery If equals are added to equals, then the wholes are equal. If equals are subtracted from equals, then the remainders are equal. Things that coincide with one another equal one another. The whole is greater than the part. HP G62-120ER Battery [edit]Parallel postulate Main article: Parallel postulate To the ancients, the parallel postulate seemed less obvious than the others. Euclid himself seems to have considered it as being qualitatively different from the others, HP G62-120ES Battery as evidenced by the organization of the Elements: the first 28 propositions he presents are those that can be proved without it. Many alternative axioms can be formulated that have the same logical consequences as the parallel postulate. For example Playfair's axiom states: In a plane, HP G62-120ET Battery through a point not on a given straight line, at most one line can be drawn that never meets the given line. Methods of proof Euclidean geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, HP G62-120EY Battery and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge.[8] In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, HP G62-120SE Batterywhich often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.[9] Strictly speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. For example a Euclidean straight line has no width, HP G62-120SL Battery but any real drawn line will. Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones—e.g., some of the Pythagoreans' proofs that involved irrational numbers, HP G62-120SS Battery which usually required a statement such as "Find the greatest common measure of ..."[10] Euclid often used proof by contradiction. Euclidean geometry also allows the method of superposition, in which a figure is transferred to another point in space. For example, proposition I.4, side-angle-side congruence of triangles, HP G62-120SW Battery is proved by moving one of the two triangles so that one of its sides coincides with the other triangle's equal side, and then proving that the other sides coincide as well. Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition.[11] HP G62-121EE Battery [edit]System of measurement and arithmetic Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, e.g., a 45-degree angle would be referred to as half of a right angle. HP G62-125EK BatteryThe distance scale is relative; one arbitrarily picks a line segment with a certain length as the unit, and other distances are expressed in relation to it. A line in Euclidean geometry is a model of the real number line. A line segment is a part of a line that is bounded by two end points, HP G62-125EL Battery and contains every point on the line between its end points. Addition is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. Measurements of area and volume are derived from distances. HP G62-125EV BatteryFor example, a rectangle with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, e.g., in the proof of book IX, HP G62-125SL Battery proposition 20. Euclid refers to a pair of lines, or a pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal, and similarly for angles. The stronger term "congruent" refers to the idea that an entire figure is the same size and shape as another figure. AlternativelyHP G62-130 Battery, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example, a 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similarHP G62-130EG Battery. [edit]Notation and terminology [edit]Naming of points and figures Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, HP G62-130EK Batteryare named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C. [edit]Complementary and supplementary angles Angles whose sum is a right angle are called complementary. HP G62-130ET Battery Complementary angles are formed when one or more rays share the same vertex and are pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays are infinite. Those whose sum is a straight angle are supplementary. HP G62-130EV Battery Supplementary angles are formed when one or more rays share the same vertex and are pointed in a direction that in between the two original rays that form the straight angle (180 degrees). The number of rays in between the two original rays are infinite like those possible in the complementary angle. HP G62-130SD Battery [edit]Modern versions of Euclid's notation In modern terminology, angles would normally be measured in degrees or radians. Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), HP G62-130SL Battery and line segments (of finite length). Euclid, rather than discussing a ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length," although he occasionally referred to "infinite lines." A "line" in Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. HP G62-134CA Battery Bridge of Asses The Bridge of Asses (Pons Asinorum) states that in isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another. HP G62-135EV Battery [12] Its name may be attributed to its frequent role as the first real test in the Elements of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross.[13] HP G62-140EL Battery [edit]Congruence of triangles Congruence of triangles is determined by specifying two sides and the angle between them (SAS), two angles and the side between them (ASA) or two angles and a corresponding adjacent side (AAS). HP G62-140EQ Battery Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). (Triangles with three equal angles are generally similar, HP G62-140ES Batterybut not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal.) [edit]Sum of the angles of a triangle acute, obtuse, and right angle limits The sum of the angles of a triangle is equal to a straight angle (180 degrees).[14] This causes an equilateral triangle to have 3 interior angles of 60 degrees. HP G62-140ET Battery Also, it causes every triangle to have at least 2 acute angles and up to 1 obtuse or right angle. [edit]Pythagorean theorem The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, HP G62-140SF Battery the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). [edit]Thales' theorem Thales' theorem, HP G62-140SS Battery named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid book I, prop 32 after the manner of Euclid book III, prop 31.[15] Tradition has it that Thales sacrificed an ox to celebrate this theorem.[16] HP G62-140US Battery [edit]Scaling of area and volume In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, , and the volume of a solid to the cube, . Euclid proved these results in various special cases such as the area of a circle[17] and the volume of a parallelepipedal solid. HP G62-143CL Battery [18] Euclid determined some, but not all, of the relevant constants of proportionality. E.g., it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder.[19] Applications Because of Euclidean geometry's fundamental status in mathematics, HP G62-144DX Battery it would be impossible to give more than a representative sampling of applications here. As suggested by the etymology of the word, one of the earliest reasons for interest in geometry was surveying,[20] and certain practical results from Euclidean geometry, such as the right-angle property of the 3-4-5 triangle, HP G62-145NR Battery were used long before they were proved formally.[21] The fundamental types of measurements in Euclidean geometry are distances and angles, and both of these quantities can be measured directly by a surveyor. Historically, distances were often measured by chains such as Gunter's chain, and angles using graduated circles and, later, HP G62-147NR Battery the theodolite. An application of Euclidean solid geometry is the determination of packing arrangements, such as the problem of finding the most efficient packing of spheres in n dimensions. This problem has applications in error detection and correction. Geometric optics uses Euclidean geometry to analyze the focusing of light by lenses and mirrors. HP G62-149WM Battery As a description of the structure of space Euclid believed that his axioms were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms, HP G62-150EE Battery [23] in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called Euclidean motions, which include translations and rotations of figures.[24] Taken as a physical description of space, HP G62-150EF Battery postulate 2 (extending a line) asserts that space does not have holes or boundaries (in other words, space is homogeneous and unbounded); postulate 4 (equality of right angles) says that space is isotropic and figures may be moved to any location while maintaining congruence; and postulate 5 (the parallel postulate) that space is flat (has no intrinsic curvature).[25] HP G62-150EQ Battery As discussed in more detail below, Einstein's theory of relativity significantly modifies this view. The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite[26] (see below) and what its topology is. HP G62-150ET BatteryModern, more rigorous reformulations of the system[27] typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1-4 are consistent with either infinite or finite space (as in elliptic geometry), HP G62-150EV Battery and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry). [edit]Later work [edit]Archimedes and Apollonius A sphere has 2/3 the volume and surface area of its circumscribing cylinder. HP G62-150SE Battery A sphere and cylinder were placed on the tomb of Archimedes at his request. Archimedes (ca. 287 BCE – ca. 212 BCE), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, HP G62-150SF Battery unlike Euclid's, is believed to have been entirely original.[28] He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers. Apollonius of Perga (ca. 262 BCE–ca. 190 BCE) is mainly known for his investigation of conic sections. HP G62-150SL Battery René Descartes. Portrait after Frans Hals, 1648. [edit]17th century: Descartes René Descartes (1596–1650) developed analytic geometry, an alternative method for formalizing geometryHP G62-153CA Battery.[29] In this approach, a point is represented by its Cartesian (x, y) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, HP G62-154CA Battery and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. The equation defining the distance between two points P = (p, q) and Q = (r, s) is then known as the Euclidean metricHP G62-165SL Battery, and other metrics define non-Euclidean geometries. In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2x + 1 (a line), or x2 + y2 = 7 (a circle). HP G62-166SB Battery Also in the 17th century, Girard Desargues, motivated by the theory of perspective, introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, projective geometry, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.[30] HP G62-200XX Battery Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge. [edit]18th centuryHP G62-201XX Battery Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763 at least 28 different proofs had been published, but all were found incorrect.[31] Leading up to this period, HP G62-219WM Battery geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those toolsHP G62-251XX Battery. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include doubling the cube and squaring the circle. In the case of doubling the cube, HP G62-400 Battery the impossibility of the construction originates from the fact that the compass and straightedge method involve first- and second-order equations, while doubling a cube requires the solution of a third-order equation. Euler discussed a generalization of Euclidean geometry called affine geometry, HP G62-450SA Battery which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, HP G62-451SA Battery and equality of length of parallel line segments (so line segments continue to have a midpoint). [edit]19th century and non-Euclidean geometry In the early 19th century, Carnot and Möbius systematically developed the use of signed angles and line segments as a way of simplifying and unifying results. HP G62-452SA Battery [32] The century's most significant development in geometry occurred when, around 1830, János Bolyai and Nikolai Ivanovich Lobachevsky separately published work on non-Euclidean geometry, in which the parallel postulate is not valid.[33] Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, HP G62-454TU Batterythe parallel postulate cannot be proved from the other postulates. In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of theorems stated in the Elements. For example, Euclid assumed implicitly that any line contains at least two points, HP G62-456TU Batterybut this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the Elements, shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex. HP G62-460TX BatteryHis axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers. Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, HP G62-467TX Battery the best known being those of Hilbert,[34] George Birkhoff,[35] and Tarski.[36] [edit]20th century and general relativity A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, HP G62-468TX Batterystars (marked with short horizontal lines) were photographed during a solar eclipse. The rays of starlight were bent by the Sun's gravity on their way to the earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry. HP G62-550EE Battery Einstein's theory of general relativity shows that the true geometry of spacetime is not Euclidean geometry.[37] For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, HP G62-a00 Batterysuch as the Earth's or the sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting the deviations from Euclidean geometry, but Einstein predicted that such deviations would exist. HP G62-a00EF Battery They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the GPS system. HP G62-a01SA Battery [38] It is possible to object to this interpretation of general relativity on the grounds that light rays might be improper physical models of Euclid's lines, or that relativity could be rephrased so as to avoid the geometrical interpretations. However, HP G62-a02SA Battery one of the consequences of Einstein's theory is that there is no possible physical test that can distinguish between a beam of light as a model of a geometrical line and any other physical model. Thus, the only logical possibilities are to accept non-Euclidean geometry as physically real, or to reject the entire notion of physical tests of the axioms of geometry, HP G62-a03SA Battery which can then be imagined as a formal system without any intrinsic real-world meaning. [edit]Treatment of infinity [edit]Infinite objects Euclid sometimes distinguished explicitly between "finite lines" (e.g., HP G62-a04EA Battery Postulate 2) and "infinite lines" (book I, proposition 12). However, he typically did not make such distinctions unless they were necessary. The postulates do not explicitly refer to infinite lines, although for example some commentators interpret postulate 3, existence of a circle with any radius, HP G62-a04SA Batteryas implying that space is infinite.[26] The notion of infinitesimally small quantities had previously been discussed extensively by the Eleatic School, but nobody had been able to put them on a firm logical basis, HP G62-a10EV Battery with paradoxes such as Zeno's paradox occurring that had not been resolved to universal satisfaction. Euclid used the method of exhaustion rather than infinitesimals.[39] Later ancient commentators such as Proclus (410–485 CE) treated many questions about infinity as issues demanding proof and, e.g., HP G62-a10SA Battery Proclus claimed to prove the infinite divisibility of a line, based on a proof by contradiction in which he considered the cases of even and odd numbers of points constituting it.[40] At the turn of the 20th century, Otto Stolz, Paul du Bois-Reymond, Giuseppe Veronese, and others produced controversial work on non-Archimedean models of Euclidean geometry, HP G62-a11SA Batteryin which the distance between two points may be infinite or infinitesimal, in the Newton–Leibniz sense.[41] Fifty years later, Abraham Robinson provided a rigorous logical foundation for Veronese's work.[42] [edit]Infinite processesHP G62-a11SE Battery One reason that the ancients treated the parallel postulate as less certain than the others is that verifying it physically would require us to inspect two lines to check that they never intersected, even at some very distant point, and this inspection could potentially take an infinite amount of time.[43] HP G62-a12SA Battery The modern formulation of proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e.g., the proof of the infinitude of primes.[44] Supposed paradoxes involving infinite series, HP G62-a12SE Batterysuch as Zeno's paradox, predated Euclid. Euclid avoided such discussions, giving, for example, the expression for the partial sums of the geometric series in IX.35 without commenting on the possibility of letting the number of terms become infinite. [edit]Logical basisHP G62-a13EE Battery This article needs attention from an expert on the subject. Please add a reason or a talk parameter to this template to explain the issue with the article. WikiProject Mathematics or the Mathematics Portal may be able to help recruit an expert. (December 2010) This section requires expansion. HP G62-a13SA Battery See also: Hilbert's axioms, Axiomatic system, and Real closed field [edit]Classical logic Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, HP G62-a13SE Battery in which every proposition is either true or false, i.e., for any proposition P, the proposition "P or not P" is automatically true. [edit]Modern standards of rigor Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries. HP G62-a14SA Battery [45] The role of primitive notions, or undefined concepts, was clearly put forward by Alessandro Padoa of the Peano delegation at the 1900 Paris conference:[45][46] ...when we begin to formulate the theory, we can imagine that the undefined symbols are completely devoid of meaning and that the unproved propositions are simply conditions imposed upon the undefined symbols. HP G62-a15EO Battery Then, the system of ideas that we have initially chosen is simply one interpretation of the undefined symbols; but..this interpretation can be ignored by the reader, who is free to replace it in his mind by another interpretation.. that satisfies the conditions... HP G62-a15SA Battery Logical questions thus become completely independent of empirical or psychological questions... The system of undefined symbols can then be regarded as the abstraction obtained from the specialized theories that result when...the system of undefined symbols is successively replaced by each of the interpretations... HP G62-a16SA Battery —Padoa, Essai d'une théorie algébrique des nombre entiers, avec une Introduction logique à une théorie déductive qulelconque That is, mathematics is context-independent knowledge within a hierarchical framework. As said by Bertrand Russell:[47] HP G62-a17EA Battery If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus, mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. HP G62-a17SA Battery —Bertrand Russell, Mathematics and the metaphysicians Such foundational approaches range between foundationalism and formalism. [edit]Axiomatic formulationsHP G62-a18SA Battery Geometry is the science of correct reasoning on incorrect figures. —George Polyá, How to Solve It, p. 208 Euclid's axioms: In his dissertation to Trinity College, Cambridge, HP G62-a19EA Battery Bertrand Russell summarized the changing role of Euclid's geometry in the minds of philosophers up to that time.[48] It was a conflict between certain knowledge, independent of experiment, and empiricism, requiring experimental input. This issue became clear as it was discovered that the parallel postulate was not necessarily valid and its applicability was an empirical matter, HP G62-a19SA Battery deciding whether the applicable geometry was Euclidean or non-Euclidean. Hilbert's axioms: Hilbert's axioms had the goal of identifying a simple and complete set of independent axioms from which the most important geometric theorems could be deduced. HP G62-a20SA BatteryThe outstanding objectives were to make Euclidean geometry rigorous (avoiding hidden assumptions) and to make clear the ramifications of the parallel postulate. Birkhoff's axioms: Birkhoff proposed four postulates for Euclidean geometry that can be confirmed experimentally with scale and protractor. HP G62-a21EA Battery [49][50][51] The notions of angle and distance become primitive concepts.[52] Tarski's axioms:Tarski (1902–1983) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basisHP G62-a21SA Battery,[53] in contrast to Hilbert's axioms, which involve point sets.[54] Tarski proved that his axiomatic formulation of elementary Euclidean geometry is consistent and complete in a certain sense: there is an algorithm that, for every proposition, can be shown either true or false. HP G62-a22SA Battery [36] (This doesn't violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.[55]) This is equivalent to the decidability of real closed fields, of which elementary Euclidean geometry is a model. [edit]Constructive approaches and pedagogyHP G62-a22SE Battery The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic. In contrast, the Greeks used construction postulates, and emphasized problem solving.[56] For the Greeks, constructions are more primitive than existence propositions, HP G62-a23SA Batteryand can be used to prove existence propositions, but not vice versa. To describe problem solving adequately requires a richer system of logical concepts.[56] The contrast in approach may be summarized:[57] Axiomatic proof: Proofs are deductive derivations of propositions from primitive premises that are ‘true’ in some sense. HP G62-a24SA Battery The aim is to justify the proposition. Analytic proof: Proofs are non-deductive derivations of hypothesis from problems. The aim is to find hypotheses capable of giving a solution to the problem. One can argue that Euclid's axioms were arrived upon in this manner. HP G62-a25EA Battery In particular, it is thought that Euclid felt the parallel postulate was forced upon him, as indicated by his reluctance to make use of it,[58] and his arrival upon it by the method of contradiction.[59] Andrei Nicholaevich Kolmogorov proposed a problem solving basis for geometry. HP G62-a25SA Battery [60][61] This work was a precursor of a modern formulation in terms of constructive type theory.[62] This development has implications for pedagogy as well.[63] If proof simply follows conviction of truth rather than contributing to its construction and is only experienced as a demonstration of something already known to be true, HP G62-a26SA Battery it is likely to remain meaningless and purposeless in the eyes of students. —Celia Hoyles, The curricular shaping of students' approach to proof Euclidean vector In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric[1] or spatial vector, HP G62-a27SA Battery [2] or – as here – simply a vector) is a geometric object that has a magnitude (or length) and direction and can be added according to the parallelogram law of addition. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, HP G62-a28SA Battery [3] and denoted by Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), HP G62-a29EA Battery their magnitude and direction can be still represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system 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 pseudovectors and tensors. HP G62-a29SA Battery It is important to distinguish Euclidean vectors from the more general concept in linear algebra of vectors as elements of a vector space. General vectors in this sense are fixed-size, ordered collections of items as in the case of Euclidean vectors, but the individual items may not be real numbers, HP G62-a30SA Batteryand the normal Euclidean concepts of length, distance and angle may not be applicable. (A vector space with a definition of these concepts is called an inner product space.) In turn, both of these definitions of vector should be distinguished from the statistical concept of a random vector. HP G62-a38EE BatteryThe individual items in a random vector are individual real-valued random variables, and are often manipulated using the same sort of mathematical vector and matrix operations that apply to the other types of vectors, but otherwise usually behave more like collections of individual values. Concepts of length, distance and angle do not normally apply to these vectors, HP G62-a40SA Battery either; rather, what links the values together is the potential correlations among them. Overview A vector is a geometric entity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction. HP G62-a43SA Battery In rigorous mathematical treatments,[4] a vector is defined as a directed line segment, or arrow, in a Euclidean space. When it becomes necessary to distinguish it from vectors as defined elsewhere, this is sometimes referred to as a geometric, spatial, or Euclidean vector. As an arrow in Euclidean space, HP G62-a44EE Batterya vector possesses a definite initial point and terminal point. Such a vector 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. HP G62-a44SA Battery Thus two arrows  and  in space represent the same free vector if they have the same magnitude and direction: that is, they are equivalent if the quadrilateral ABB′A′ is a parallelogram. If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. HP G62-a45SA Battery The term vector also has generalizations to higher dimensions and to more formal approaches with much wider applications. [edit]Examples in one dimension Since the physicist's concept of force has a direction and a magnitude, HP G62-a50SG Battery it may be seen as a vector. As an example, consider a rightward force F of 15 newtons. If the positive axis 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, HP G62-a53SG Batterythe vector representation of a displacement Δs of 4 meters to the right would be 4 m or −4 m, and its magnitude would be 4 m regardless. [edit]In physics and engineering Vectors are fundamental in the physical sciences. HP G62-a60SA BatteryThey can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (0,5) (in 2 dimensions with the positive y axis as 'up'). HP G62-b00SA BatteryAnother quantity represented by a vector is force, since it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement, acceleration, momentum, and angular momentum. Other physical vectors, such as the electric and magnetic field, HP G62-b09SA Batteryare represented as a system of vectors at each point of a physical space; that is, a vector field. [edit]In Cartesian space In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point. HP G62-b10SA BatteryFor instance, the points A = (1,0,0) and B = (0,1,0) in space determine the free vector  pointing from the point x=1 on the x-axis to the point y=1 on the y-axis. Typically in Cartesian coordinates, one considers primarily bound vectors. A bound vector is determined by the coordinates of the terminal point, HP G62-b11SA Batteryits initial point always having the coordinates of the origin O = (0,0,0). Thus the bound vector represented by (1,0,0) is a vector of unit length pointing from the origin up the positive x-axis. The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. HP G62-b12SA Battery For example, the sum of the vectors (1,2,3) and (−2,0,4) is the vector (1, 2, 3) + (−2, 0, 4) = (1 − 2, 2 + 0, 3 + 4) = (−1, 2, 7). [edit]Euclidean and affine vectors In the geometrical and physical settings, sometimes it is possible to associate, in a natural way, HP G62-b13EA Battery a length or magnitude and a direction to vectors. In turn, the notion of direction is strictly associated with the notion of an angle between two vectors. When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length (the square root of the dot product of a vector by itself) and angle (a function of the dot product between any two vectors). HP G62-b13SA BatteryIn three dimensions, it is further possible to define a cross product which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of the parallelogram). However, HP G62-b14SA Battery it is not always possible or desirable to define the length of a vector in a natural way. This more general type of spatial vector is the subject of vector spaces (for bound vectors) and affine spaces (for free vectors). An important example is Minkowski space that is important to our understanding of special relativity, HP G62-b15SA Battery where there is a generalization of length that permits non-zero vectors to have zero length. Other physical examples come from thermodynamics, where many of the quantities of interest can be considered vectors in a space with no notion of length or angle.[5] [edit]GeneralizationsHP G62-b16EA Battery In physics, as well as mathematics, a vector is often identified with a tuple, or list of numbers, which depend on some auxiliary coordinate system or reference frame. When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform. HP G62-b16SA BatteryThe vector itself has not changed, but the reference frame has, so the components of the vector (or measurements taken with respect to the reference frame) 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 coordinates. HP G62-b17EO Battery 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. If you change units (a special case of a change of coordinates) from meters to milimetersHP G62-b17SA Battery, 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. See covariance and contravariance of vectors. Tensors are another type of quantity that behave in this way; in fact a vector is a special type of tensor. HP G62-b18SA Battery In pure mathematics, a vector is any element of a vector space over some field 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. HP G62-b19SA BatteryContravariance captures the physical intuition behind the idea that a vector has "magnitude and direction". [edit]History This section requires expansion. The concept of vector, HP G62-b20SA Battery as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions.[6] The immediate predecessor of vectors were quaternions, devised by William Rowan Hamilton in 1843 as a generalization of complex numbersHP G62-B20so Battery. His search was for a formalism to enable the analysis of three-dimensional space in the same way that complex numbers had enabled analysis of two-dimensional space. In 1846 Hamilton divided his quaternions into the sum of real and imaginary parts that he respectively called "scalar" and "vector":HP G62-b21SA Battery 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. HP G62-b22SA Battery —W. R. Hamilton, London, Edinburgh & Dublin Philososphical Magazine 3rd series 29 27 (1846) Whereas complex numbers have one number  whose square is negative one, quaternions have three independent such numbersHP G62-b23SA Battery . Multiplication of these numbers by each other is not commutative, e.g., . Multiplication of two quaternions yields a third quaternion whose scalar part is the negative of the modern dot product and whose vector part is the modern cross product. Peter Guthrie Tait carried the quaternion standard after Hamilton. HP G62-b24SA Battery His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator and is very close to modern vector analysis. Josiah Willard Gibbs, HP G62-b25SA Battery who was exposed to quaternions through James Clerk Maxwell'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.[6] [edit]RepresentationsHP G62-b26SA Battery Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, as a. (Uppercase letters are typically used to represent matrices.) Other conventions include  or a, especially in handwriting. HP G62-b27EA BatteryAlternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, which is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as  or AB. Vectors are usually shown in graphs or other diagrams as arrows (directed line segments), as illustrated in the figure. HP G62-b27SA Battery Here the point A is called the origin, tail, base, or initial point; 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, while the direction in which the arrow points indicates the vector's direction. On a two-dimensional diagram, HP G62 Batterysometimes a vector perpendicular to the plane of the diagram is 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. HP G62 Notebook PC Series Battery These can be thought of as viewing the tip of an arrow head on and viewing the vanes of an arrow from the back. A vector in the Cartesian plane, showing the position of a point A with coordinates (2,3). 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. HP G62t-100 CTO BatteryThe endpoint of a vector can be identified with an ordered list of n real numbers (n-tuple). These numbers are the coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system, and are typically called the scalar components (or scalar projections) of the vector on the axes of the coordinate system. HP G62t Battery 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 The notion that the tail of the vector coincides with the origin is implicit and easily understood. HP G72-100 BatteryThus, the more explicit notation  is usually not deemed necessary and very rarely used. In three dimensional Euclidean space (or ), vectors are identified with triples of scalar components: also writtenHP G72-101SA Battery These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows: Another way to represent a vector in n-dimensions is to introduce the standard basis vectors. For instance, HP G72-102SA Battery in three dimensions, there are three of them: These have the intuitive interpretation as vectors of unit length pointing up the x, y, and z axis of a Cartesian coordinate system, respectively, and they are sometimes referred to as versors of those axes. In terms of these, any vector a in  can be expressed in the form: orHP G72-105SA Battery where a1, a2, a3 are called the vector components (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x, y, and z (see figure), while a1, a2, a3 are the respective scalar components (or scalar projections). In introductory physics textbooks, HP G72-110EL Batterythe standard basis vectors are often instead denoted  (or , in which the hat symbol ^ typically denotes unit vectors). In this case, the scalar and vector components are denoted ax, ay, az, and ax, ay, az. Thus, The notation ei is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, HP G72-110EV Batteryand engineering. [edit]Decomposition As explained above a vector is often described by a set of vector components that are mutually perpendicular and add up to form the given vector. HP G72-110SA BatteryTypically, these components are the projections of the vector on a set of reference axes (or basis vectors). The vector is said to be decomposed or resolved with respect to that set. Illustration of tangential and normal components of a vector to a surface. However, the decomposition of a vector into components is not unique, HP G72-110SD Batterybecause it depends on the choice of the axes on which the vector is projected. Moreover, the use of Cartesian versors such as  as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of the versors of a Cylindrical coordinate system () or Spherical coordinate system ().HP G72-110SO Battery The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry respectively. The choice of a coordinate system doesn't affect the properties of a vector or its behaviour under transformations. HP G72-110SW Battery A vector can be also decomposed with respect to "non-fixed" axes which change their orientation 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, HP G72-120EG Batterythe radial and tangential components of a vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it.[7] In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., HP G72-120EP Batterya global coordinate system, or inertial reference frame). [edit]Basic properties The following section uses the Cartesian coordinate system with basis vectors and assumes that all vectors have the origin as a common base point. A vector a will be written asHP G72-120EV Battery [edit]Equality Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors andHP G72-120EW Battery are equal if [edit]Addition and subtraction Assume now that a and b are not necessarily equal vectors, but that they may have different magnitudes and directions. HP G72-120SD Battery The sum of a and b is The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below: HP G72-120SG Battery This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors that have the same base point, it 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). HP G72-120SO Battery The difference of a and b is Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the end points of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. HP G72-130 BatteryThat arrow represents the vector a − b, as illustrated below: [edit]Scalar multiplication Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications 2a and −a of a vector a A vector may also be multiplied, or re-scaled, HP G72-130ED Battery by a real number 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 Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. HP G72-130EG Battery 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: HP G72-130EV Battery Scalar multiplication is distributive over vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a − b = a + (−1)b. [edit]Length The length or magnitude or norm of the vector a is denoted by ||a|| or, HP G72-130SA Battery less commonly, |a|, which is not to be confused with the absolute value (a scalar "norm"). The length of the vector a can be computed with the Euclidean norm which is a consequence of the Pythagorean theorem since the basis vectors e1, e2, e3 are orthogonal unit vectors. HP G72-130SF Battery This happens to be equal to the square root of the dot product, discussed below, of the vector with itself: Unit vector The normalization of a vector a into a unit vector â Main article: Unit vectorHP G72-140ED Battery 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 âHP G72-150EF Battery. To normalize a vector a = [a1, a2, a3], scale the vector by the reciprocal of its length ||a||. That is: Null vector Main article: Null vectorHP G72-251XX Battery The null vector (or zero vector) is the vector with length zero. Written out in coordinates, the vector is (0,0,0), and it is commonly denoted , or 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 which is a multiple of the null vector). HP G72-260US BatteryThe sum of the null vector with any vector a is a (that is, 0+a=a). [edit]Dot product Main article: dot product The dot product of two vectors a and b (sometimes called the inner product, HP G72-a10SA Batteryor, since its result is a scalar, the scalar product) is denoted by a ∙ b and is defined as: where θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a. HP G72-a20SA Battery The dot product can also be defined as the sum of the products of the components of each vector as [edit]Cross product Main article: Cross productHP G72-a30SA Battery The cross product (also called the vector product or outer product) is only meaningful in three or seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. HP G72-a40SA BatteryThe cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b which completes a right-handed system. The right-handedness constraint is necessary because there exist two unit vectors that are perpendicular to both a and b, HP G72-b01EA Batterynamely, n and (–n). An illustration of the cross product The cross product a × b is defined so that a, b, and a × b also becomes a right-handed system (but note that a and b are not necessarily orthogonal). This is the right-hand rule. HP G72-b01SA Battery 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 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 instead of a vector (see below). HP G72-b02SA Battery [edit]Scalar triple product Main article: Scalar triple product The scalar triple product (also called the box product or mixed triple product) is not really a new operator, HP G72-b10SA Battery 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: It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. HP G72-b15SA BatterySecond, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are right-handed. HP G72-b20SA Battery In components (with respect to a right-handed orthonormal basis), if the three vectors are thought of as rows (or columns, but in the same order), the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows The scalar triple product is linear in all three entries and anti-symmetric in the following sense: HP G72 Battery [edit]Multiple Cartesian bases All examples thus far have dealt with vectors expressed in terms of the same basis, namely, 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, HP G72 Notebook PC Series Batteryand still remain the same vector. For example, using the vector a from above, where n1,n2,n3 form another orthonormal basis not aligned with e1,e2,e3. The values of u, v, and w are such that the resulting vector sum is exactly a. It is not uncommon to encounter vectors known in terms of different bases (for example, HP G72T-200 CTO Batteryone basis fixed to the Earth and a second basis fixed to a moving vehicle). In order to perform many of the operations defined above, it is necessary to know the vectors in terms of the same basis. One simple way to express a vector known in one basis in terms of another uses column matrices that represent the vector in each basis along with a third matrix containing the information that relates the two bases. HP G72t BatteryFor example, in order to find the values of u, v, and w that define a in the n1,n2,n3 basis, a matrix multiplication may be employed in the form where each matrix element cjk is the direction cosine relating nj to ek.[8] The term direction cosine refers to the cosine of the angle between two unit vectors, which is also equal to their dot product. HP Pavilion dv6-3000 Battery [8] By referring collectively to e1,e2,e3 as the e basis and to n1,n2,n3 as the n basis, the matrix containing all the cjk is known as the "transformation matrix from e to n", or the "rotation matrix from e to n" (because it can be imagined as the "rotation" of a vector from one basis to another), HP Pavilion dv6-3005sa Battery or the "direction cosine matrix from e to n"[8] (because it contains direction cosines). The properties of a rotation matrix are such that its inverse is equal to its transpose. This means that the "rotation matrix from e to n" is the transpose of "rotation matrix from n to e".HP Pavilion dv6-3005TX Battery 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.[8] [edit]Other dimensions With the exception of the cross and triple products, the above formula generalise to two dimensions and higher dimensions. HP Pavilion dv6-3006TX BatteryFor example, addition generalises to two dimensions the addition of and in four dimension The cross product generalises to the exterior product, whose result is a bivector, which in general is not a vector. HP Pavilion dv6-3010sa Battery In two dimensions this is simply a scalar The seven-dimensional cross product is similar to the cross product in that its result is a seven-dimensional vector orthogonal to the two arguments. [edit]PhysicsHP Pavilion dv6-3011TX Battery Vectors have many uses in physics and other sciences. [edit]Length and units In abstract vector spaces, the length of the arrow depends on a dimensionless scale. If it represents, for example, HP Pavilion dv6-3015sa Battery 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 1:250 and 1 m:50 N respectively. HP Pavilion dv6-3020sa Battery Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance. [edit]Vector-valued functionsHP Pavilion dv6-3025sa Battery Main article: Vector-valued function 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, HP Pavilion dv6-3026tx Battery 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 from calculus continue to hold for the derivative and integral of vector-valued functions. [edit]Position, velocity and accelerationHP Pavilion dv6-3030sa Battery The position of a point x=(x1, x2, x3) in three dimensional space can be represented as a position vector whose base point is the origin The position vector has dimensions of length. HP Pavilion dv6-3030TX Battery Given two points x=(x1, x2, x3), y=(y1, y2, y3) their displacement is a vector 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 v of a point or particle is a vector, HP Pavilion dv6-3031sa Batteryits length gives the speed. For constant velocity the position at time t will be where x0 is the position at time t=0. Velocity is the time derivative of position. Its dimensions are length/time. Acceleration a of a point is vector which is the time derivative of velocity. HP Pavilion dv6-3032sa BatteryIts dimensions are length/time2. [edit]Force, energy, work Force is a vector with dimensions of mass×length/time2 and Newton's second law is the scalar multiplication Work is the dot product of force and displacementHP Pavilion dv6-3032TX Battery [edit]Vectors as directional derivatives A vector may also be defined as a directional derivative: consider a function  and a curve . Then the directional derivative of  is a scalar defined asHP Pavilion dv6-3033sa Battery where the index  is summed over the appropriate number of dimensions (for example, from 1 to 3 in 3-dimensional Euclidean space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to : HP Pavilion dv6-3035sa Battery The directional derivative can be rewritten in differential form (without a given function ) as Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. A vector can therefore be defined precisely asHP Pavilion dv6-3040sa Battery [edit]Vectors, pseudovectors, and transformations An alternative characterization of Euclidean vectors, especially in physics, describes them as lists of quantities which behave in a certain way under a coordinate transformation. HP Pavilion dv6-3042TX BatteryA contravariant vector is required to have components that "transform like the coordinates" under changes of coordinates such as rotation and dilation. The vector itself does not change under these operations; instead, the components of the vector make a change that cancels the change in the spatial axes, in the same way that co-ordinates change. HP Pavilion dv6-3044sa BatteryIn other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, like the co-ordinates, would reduce in an exactly compensating way. HP Pavilion dv6-3045sa BatteryMathematically, if the coordinate system undergoes a transformation described by an invertible matrix M, so that a coordinate vector x is transformed to x′ = Mx, then a contravariant vector v must be similarly transformed via v′ = Mv. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. HP Pavilion dv6-3046sa Battery For example, if v consists of the x, y, and z-components of velocity, then v is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an abstract vector, HP Pavilion dv6-3047sa Batterybut this vector would not be contravariant, since rotating the box does not change the box's length, width, and height. Examples of contravariant vectors include displacement, velocity, electric field, momentum, force, and acceleration. HP Pavilion dv6-3048sa Battery In the language of differential geometry, the requirement that the components of a vector transform according to the same matrix of the coordinate transition is equivalent to defining a contravariant vector to be a tensor of contravariant rank one. Alternatively, a contravariant vector is defined to be a tangent vector, HP Pavilion dv6-3048tx Battery and the rules for transforming a contravariant vector follow from the chain rule. Some vectors transform like contravariant vectors, except that when they are reflected through a mirror, they flip and gain a minus sign. HP Pavilion dv6-3050eo BatteryA transformation that switches right-handedness to left-handedness and vice versa like a mirror does is said to change the orientation of space. A vector which gains a minus sign when the orientation of space changes is called a pseudovector or an axial vector. Ordinary vectors are sometimes called true vectors or polar vectors to distinguish them from pseudovectors. HP Pavilion dv6-3050sa Battery Pseudovectors occur most frequently as the cross product of two ordinary vectors. One example of a pseudovector is angular velocity. Driving in a car, and looking forward, each of the wheels has an angular velocity vector pointing to the left. If the world is reflected in a mirror which switches the left and right side of the car, HP Pavilion dv6-3055sa Batterythe reflection of this angular velocity vector points to the right, but the actual angular velocity vector of the wheel still points to the left, corresponding to the minus sign. Other examples of pseudovectors include magnetic field, torque, or more generally any cross product of two (true) vectors. This distinction between vectors and pseudovectors is often ignored, HP Pavilion dv6-3056sa Battery but it becomes important in studying symmetry properties. See parity (physics).