194:
730:
1043:
841:, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are
1241:. Vector-valued functions, where the output is a vector, are scrutinized using calculus to derive essential insights into motion within three-dimensional space. Vector calculus extends traditional calculus principles to vector fields, introducing operations like
331:
means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of
169:
is generally not used for elements of these vector spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).
1910:
315:
406:
acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example,
1233:
serves as a foundational mathematical tool in the realm of vectors, offering a framework for the analysis and manipulation of vector quantities in diverse scientific disciplines, notably
533:, a physical vector may be endowed with additional structure compared to a geometrical vector. A bound vector is defined as the combination of an ordinary vector quantity and a
1122:
1518:, or cross product, an operation on two vectors in a three-dimensional Euclidean space, producing a third three-dimensional Euclidean vector perpendicular to the original two
1498:
561:
and an accompanying two-dimensional position vector in meters, for a total of four numbers on the plane (and six in space). A simpler example of a bound vector is the
414:), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the
1466:
1140:, even in contexts where vector-space operations do not apply. More generally, when some data can be represented naturally by vectors, they are often called
2169:
2106:
2019:
1980:
1947:
1920:
1884:
1060:
1426:, first published in 1901, which did much to standardize the notation and vocabulary of three-dimensional linear algebra and vector calculus
122:
Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a
138:
generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called a
2150:
2083:
1726:
1082:
583:, over which the vector quantity can be translated (without rotations). A free vector is a vector quantity having an undefined
1417:
1269:, essential for computations involving scalar or vector fields over three-dimensional regions, contribute to understanding
2127:
594:
Aside from the notion of units and support, physical vector quantities may also differ from
Euclidean vectors in terms of
1064:
1053:
1184:
1144:
even when addition and scalar multiplication of vectors are not valid operations on these data. Here are some examples.
936:
is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called
878:
2122:
1548:
831:
281:
1432:, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
1165:, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
830:, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying
1845:
1820:
1677:
1365:
1371:
1337:
904:
838:
783:. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars:
1682:
1386:
1331:
1286:
992:
in the space of rotation vectors may induce a curve in the space of rotations that is not a loop. Also, the
815:
333:
225:
143:
73:
31:
1557:, an econometric model used to capture the evolution and the interdependencies between multiple time series
1539:
1535:
1354:
884:
819:
671:
562:
224:
or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has
139:
1554:
1133:
776:
607:
584:
545:
507:
411:
260:
255:
131:
1587:, a solitary wave with multiple components coupled together that maintains its shape during propagation
865:
and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example,
549:, with a definite initial point besides the magnitude and direction of the main vector. For example, a
1098:
573:
and unit (length an meters). A sliding vector is the combination of an ordinary vector quantity and a
1423:
1377:
1315:
933:
888:
827:
770:
689:
603:
535:
492:
251:
233:
158:
1578:
1441:
1381:
1304:
1298:
1156:
1011:
997:
912:
908:
896:
792:
667:
588:
575:
468:
419:
345:
1937:
1667:
1653:
1476:
1321:
1250:
1174:
1032:
1015:
870:
858:
570:
462:(also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued
384:
vectors as an example of the more generalized concept of vectors defined simply as elements of a
1970:
1874:
1569:, a function defined on a family of sets and taking vector values satisfying certain properties
2146:
2102:
2079:
2015:
1976:
1943:
1916:
1880:
1722:
1521:
1270:
1218:
944:
803:
762:
675:
504:
463:
415:
407:
247:
123:
50:
1690:
1451:
1171:, in musical set theory, an array that expresses the intervallic content of a pitch-class set
2117:
2007:
1590:
1470:
1392:
1258:
1152:
1132:
real numbers has a natural structure of vector space defined by component-wise addition and
1001:
1000:, while the manifold of rotations is not. Spinors are elements of a vector subspace of some
967:
799:
636:
632:
484:
455:
365:
221:
186:
179:
85:
54:
1509:
1503:
1435:
1266:
1208:
1196:
1168:
1148:
981:
892:
866:
788:
685:
647:
643:
595:
554:
496:
475:
459:
447:
440:
381:
242:
154:
150:
127:
116:
1438:, a branch of mathematics concerned with differentiation and integration of vector fields
1742:
2139:
1584:
1566:
1515:
1411:
1274:
1214:
1162:
874:
850:
823:
628:
500:
377:
368:
have close analogues for vectors, operations which obey the familiar algebraic laws of
361:
229:
162:
2011:
2163:
1429:
1310:
1254:
1180:
1019:
916:
698:
693:
373:
369:
2066:
741:(red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum
17:
2134:
1596:
1572:
1560:
1350:
989:
920:
784:
758:
722:
715:
680:
659:
566:
565:
vector from an initial point to an end point; in this case, the bound vector is an
423:
399:
385:
237:
58:
1766:
883:
Many vector spaces that are considered in mathematics are also endowed with other
193:
1398:
1334:, a vector space V equipped with a non-degenerate, skew-symmetric, bilinear form
1238:
1192:
1007:
954:
900:
750:
655:
526:
488:
357:
349:
217:
209:
101:
38:
1340:, a blend of topological structure with the algebraic concept of a vector space
729:
1246:
1200:
1188:
1010:, an infinite sequence of elements of a commutative ring, which belongs to an
948:
842:
663:
587:
or region of application; it can be freely translated with no consequences; a
422:
and transform in a similar way under changes of the coordinate system include
2073:
2098:
1791:
1401:, a set of closely related concepts of the flow determined by a vector field
599:
530:
69:
569:
of points in the same position space, with all coordinates having the same
2038:
1543:
1358:
1242:
1230:
993:
958:
862:
811:
702:
651:
611:
480:
395:
353:
89:
81:
65:
678:). In the natural sciences, the term "vector quantity" also encompasses
264:. A vector is frequently depicted graphically as an arrow connecting an
1301:, a type of vector space that includes the extra structure of gradation
1234:
754:
558:
522:
511:
391:
213:
42:
1551:, a linear transformation that converts a matrix into a column vector
1257:, crucial for calculating work along a path within force fields, and
973:
515:
427:
72:) for quantities that have both a magnitude and a direction, such as
1177:, in statistics, a vector with non-negative entries that sum to one.
769:, can be added together and multiplied ("scaled") by numbers called
1698:
1265:, illustrate the practical utility of calculus in vector analysis.
1912:
Mechanical
Systems, Classical Models: Volume 1: Particle Mechanics
1125:
807:
728:
550:
403:
192:
135:
112:
77:
1532:, a linear mapping producing a vector parallel to a second vector
1448:, a vector differential operator represented by the nabla symbol
1368:, a vector field that is the gradient of a scalar potential field
1939:
Analytical
Mechanics: Solutions to Problems in Classical Physics
1262:
97:
93:
2039:"ISO 80000-2:2019 - Quantities and units - Part 2: Mathematics"
1374:, a vector field defined for any energy function or Hamiltonian
1253:, which find applications in physics and engineering contexts.
1036:
2002:"Appendix A. Linear Algebra from a Geometric Point of View".
1975:. Dover Books on Mathematics. Dover Publications. p. 2.
1904:
1902:
1876:
Dynamics of
Particles and Rigid Bodies: A Systematic Approach
418:
used to describe it. Other vector-like objects that describe
61:. They have to be expressed by both magnitude and direction.
1357:
that, generally, has a domain of the same dimension (as a
149:
Many vector spaces are considered in mathematics, such as
2006:. Ithaca, NY: David W. Henderson. 2013. p. 121–138.
1699:"Earliest Known Uses of Some of the Words of Mathematics"
1512:, a type of differential operator used in vector calculus
610:
of meters: it includes a position
Euclidean vector and a
1969:
Borisenko, A.I.; Tarapov, I.E.; Silverman, R.A. (2012).
1500:, is a differential operator defined over a vector field
2075:
Introduction to Tensor
Calculus and Continuum Mechanics
2033:
2031:
1868:
1866:
791:. Scalars can also be, more generally, elements of any
1815:
1813:
1811:
861:. Finite-dimensional vector spaces occur naturally in
284:
236:. Euclidean vectors can be added and scaled to form a
1821:"Details for IEV number 102-03-21: "vector quantity""
1660:= "I carry". For historical development of the word
1479:
1454:
1101:
1063:. Please help to ensure that disputed statements are
984:. In fact, rotation vectors represent well rotations
1581:, a quantization technique used in signal processing
733:
Vector addition and scalar multiplication: a vector
1395:, a vector field whose curl is a given vector field
980:, have been introduced for extending the notion of
2138:
1666:
1492:
1460:
1116:
822:. The concept of vector spaces is fundamental for
309:
142:, and a vector space formed by tuples is called a
119:(of numbers or other objects) of a fixed length.
1563:, a boson with the spin quantum number equal to 1
466:. It is typically formulated as the product of a
1721:(2nd. ed.). London: Clarendon Press. 2001.
1506:, common notation used when working with vectors
705:are also admitted as physical vector quantities.
319:A vector is what is needed to "carry" the point
310:{\textstyle {\stackrel {\longrightarrow }{AB}}.}
2004:Differential Geometry: A Geometric Introduction
543:. Bound vector quantities are formulated as a
380:. These operations and associated laws qualify
53:that cannot be expressed by a single number (a
1318:, a vector space equipped with a partial order
873:infinite-dimensional vector spaces, and many
8:
1972:Vector and Tensor Analysis with Applications
1846:"Details for IEV number 102-03-04: "vector""
1307:, a vector space on which a norm is defined
1159:and magnitude is the angle of the rotation.
670:, at a particular instant, of a continuous
557:has two Cartesian components in SI unit of
1626:
1575:, a meson with total spin 1 and odd parity
1473:, the vector Laplace operator, denoted by
779:must satisfy certain requirements, called
642:Vector quantities are a generalization of
591:is a prototypical example of free vector.
2067:Vectors - The Feynman Lectures on Physics
1915:. Springer Science & Business Media.
1879:. Cambridge University Press. p. 3.
1850:International Electrotechnical Vocabulary
1825:International Electrotechnical Vocabulary
1484:
1478:
1453:
1155:whose direction is that of the axis of a
1108:
1104:
1103:
1100:
1083:Learn how and when to remove this message
837:Vector spaces are characterized by their
650:. Individual vectors may be ordered in a
296:
288:
286:
285:
283:
64:Historically, vectors were introduced in
1261:, employed to determine quantities like
1211:that takes its values in a vector space.
1059:Relevant discussion may be found on the
775:. The operations of vector addition and
1608:
1014:, and has been introduced for handling
2095:Encyclopedic Dictionary of Mathematics
1615:
1641:
1389:, a vector field with zero divergence
84:. Such quantities are represented by
7:
1691:participating institution membership
1136:. It is common to call these tuples
940:, mainly due to historical reasons.
666:. A vector may also result from the
111:is also used, in some contexts, for
1909:Teodorescu, Petre P. (2007-06-06).
1637:
1481:
1455:
761:(also called a linear space) is a
737:(blue) is added to another vector
646:and can be further generalized as
390:Vectors play an important role in
25:
2170:Vectors (mathematics and physics)
1743:"vector | Definition & Facts"
1422:a textbook on vector calculus by
1293:Vector spaces with more structure
1018:propagation in the operations on
907:, which include function spaces,
2141:Geometry: A comprehensive course
1117:{\displaystyle \mathbb {R} ^{n}}
1041:
721:This section is an excerpt from
446:This section is an excerpt from
185:This section is an excerpt from
2012:10.3792/euclid/9781429799843-13
826:, together with the concept of
1936:Merches, I.; Radu, D. (2014).
988:, but not globally, because a
297:
1:
1719:The Oxford English Dictionary
1542:that has a vector space as a
765:whose elements, often called
174:Vectors in Euclidean geometry
1185:multivariate random variable
879:cardinality of the continuum
658:), such as position vectors
2123:Encyclopedia of Mathematics
1593:, a type of audio synthesis
1549:Vectorization (mathematics)
1493:{\displaystyle \nabla ^{2}}
1027:Data represented by vectors
903:. This is also the case of
832:systems of linear equations
598:. For example, an event in
402:of a moving object and the
2186:
2072:Heinbockel, J. H. (2001).
1942:. CRC Press. p. 379.
1030:
857:, and its dimension is an
802:, which allow modeling of
720:
713:
635:is adopted instead of the
445:
438:
184:
177:
68:and physics (typically in
57:), or to elements of some
29:
1678:Oxford English Dictionary
1366:Conservative vector field
1217:, a vector of 0s and 1s (
905:topological vector spaces
798:Vector spaces generalize
49:is a term that refers to
27:Element of a vector space
1372:Hamiltonian vector field
1338:Topological vector space
1277:, and fluid flow rates.
602:may be represented as a
2078:. Trafford Publishing.
1747:Encyclopedia Britannica
1683:Oxford University Press
1656:of vehere, "to carry"/
1461:{\displaystyle \nabla }
1387:Solenoidal vector field
1332:Symplectic vector space
1287:Vector (disambiguation)
996:of rotation vectors is
814:, that have not only a
197:A vector pointing from
144:coordinate vector space
2116:Ivanov, A.B. (2001) ,
1536:Vector-valued function
1494:
1462:
1380:, a vector field on a
1355:vector-valued function
1118:
1012:algebra over this ring
887:. This is the case of
849:if its dimension is a
746:
696:over Earth's surface.
672:vector-valued function
311:
205:
140:Euclidean vector space
1796:mathworld.wolfram.com
1555:Vector autoregression
1495:
1463:
1134:scalar multiplication
1119:
951:with a zero real part
845:). A vector space is
777:scalar multiplication
732:
688:or three-dimensional
631:). In that case, the
619: ⋅
608:coherent derived unit
546:directed line segment
508:Cartesian coordinates
312:
261:directed line segment
196:
132:scalar multiplication
2093:ItĂ´, Kiyosi (1993).
1477:
1452:
1378:Killing vector field
1328:-graded vector space
1316:Ordered vector space
1207:may also refer to a
1099:
1052:factual accuracy is
966:, an element of the
934:algebra over a field
909:inner product spaces
897:associative algebras
895:, polynomial rings,
855:infinite-dimensional
604:position four-vector
536:point of application
503:may be expressed as
346:algebraic operations
282:
252:units of measurement
30:For other uses, see
18:Vector (mathematics)
1790:Weisstein, Eric W.
1681:(Online ed.).
1579:Vector quantization
1442:Vector differential
1382:Riemannian manifold
1361:) as its codomain,
1305:Normed vector space
1299:Graded vector space
1226:Vectors in calculus
853:. Otherwise, it is
804:physical quantities
589:displacement vector
576:line of application
469:unit of measurement
420:physical quantities
246:is a vector-valued
100:are represented by
88:in the same way as
1771:www.mathsisfun.com
1654:perfect participle
1490:
1458:
1322:Super vector space
1175:Probability vector
1114:
1033:Vector (data type)
970:of a vector space.
928:Vectors in algebra
847:finite-dimensional
747:
692:of space, such as
571:quantity dimension
529:, particularly in
307:
258:, formulated as a
206:
134:that satisfy some
2108:978-0-262-59020-4
2021:978-1-4297-9984-3
1982:978-0-486-13190-0
1949:978-1-4822-3940-9
1922:978-1-4020-5442-6
1886:978-0-521-85811-3
1689:(Subscription or
1522:Vector projection
1271:mass distribution
1259:surface integrals
1093:
1092:
1085:
945:Vector quaternion
859:infinite cardinal
800:Euclidean vectors
676:pendulum equation
648:tensor quantities
644:scalar quantities
495:. For example, a
464:physical quantity
435:Vector quantities
416:coordinate system
327:; the Latin word
301:
278:, and denoted by
248:physical quantity
86:geometric vectors
16:(Redirected from
2177:
2156:
2144:
2130:
2112:
2097:(2nd ed.).
2089:
2054:
2053:
2051:
2050:
2035:
2026:
2025:
1999:
1993:
1992:
1990:
1989:
1966:
1960:
1959:
1957:
1956:
1933:
1927:
1926:
1906:
1897:
1896:
1894:
1893:
1873:Rao, A. (2006).
1870:
1861:
1860:
1858:
1857:
1842:
1836:
1835:
1833:
1832:
1817:
1806:
1805:
1803:
1802:
1787:
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1778:
1777:
1763:
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1753:
1739:
1733:
1732:
1715:
1709:
1708:
1706:
1705:
1694:
1686:
1674:
1650:
1644:
1640:, p. 1678;
1635:
1629:
1624:
1618:
1613:
1591:Vector synthesis
1530:vector component
1524:, also known as
1499:
1497:
1496:
1491:
1489:
1488:
1471:Vector Laplacian
1467:
1465:
1464:
1459:
1393:Vector potential
1267:Volume integrals
1197:random variables
1153:Euclidean vector
1131:
1123:
1121:
1120:
1115:
1113:
1112:
1107:
1088:
1081:
1077:
1074:
1068:
1065:reliably sourced
1045:
1044:
1037:
1002:Clifford algebra
968:exterior algebra
963:
893:field extensions
891:, which include
881:as a dimension.
867:polynomial rings
744:
740:
736:
637:Euclidean metric
633:Minkowski metric
626:
485:Euclidean vector
456:natural sciences
316:
314:
313:
308:
303:
302:
300:
295:
287:
222:Euclidean vector
187:Euclidean vector
180:Euclidean vector
155:polynomial rings
151:extension fields
126:equipped with a
117:finite sequences
21:
2185:
2184:
2180:
2179:
2178:
2176:
2175:
2174:
2160:
2159:
2153:
2133:
2115:
2109:
2092:
2086:
2071:
2063:
2058:
2057:
2048:
2046:
2037:
2036:
2029:
2022:
2001:
2000:
1996:
1987:
1985:
1983:
1968:
1967:
1963:
1954:
1952:
1950:
1935:
1934:
1930:
1923:
1908:
1907:
1900:
1891:
1889:
1887:
1872:
1871:
1864:
1855:
1853:
1844:
1843:
1839:
1830:
1828:
1819:
1818:
1809:
1800:
1798:
1789:
1788:
1784:
1775:
1773:
1765:
1764:
1760:
1751:
1749:
1741:
1740:
1736:
1729:
1717:
1716:
1712:
1703:
1701:
1696:
1688:
1665:
1652:Latin: vectus,
1651:
1647:
1636:
1632:
1627:Heinbockel 2001
1625:
1621:
1614:
1610:
1605:
1526:vector resolute
1510:Vector operator
1504:Vector notation
1480:
1475:
1474:
1450:
1449:
1436:Vector calculus
1418:Vector Analysis
1408:
1347:
1327:
1295:
1283:
1228:
1209:random variable
1169:Interval vector
1149:Rotation vector
1129:
1102:
1097:
1096:
1089:
1078:
1072:
1069:
1058:
1050:This section's
1046:
1042:
1035:
1029:
982:rotation vector
959:
930:
925:
924:
875:function spaces
789:complex numbers
742:
738:
734:
726:
718:
712:
707:
706:
684:defined over a
627:(involving the
625:
615:
555:Euclidean plane
541:point of action
497:position vector
476:numerical value
460:vector quantity
451:
448:Vector quantity
443:
441:Vector quantity
437:
432:
431:
280:
279:
254:and possibly a
243:vector quantity
190:
182:
176:
163:function spaces
128:vector addition
35:
28:
23:
22:
15:
12:
11:
5:
2183:
2181:
2173:
2172:
2162:
2161:
2158:
2157:
2151:
2131:
2113:
2107:
2090:
2084:
2069:
2062:
2059:
2056:
2055:
2027:
2020:
1994:
1981:
1961:
1948:
1928:
1921:
1898:
1885:
1862:
1837:
1807:
1782:
1758:
1734:
1727:
1710:
1645:
1630:
1619:
1607:
1606:
1604:
1601:
1600:
1599:
1594:
1588:
1585:Vector soliton
1582:
1576:
1570:
1567:Vector measure
1564:
1558:
1552:
1546:
1533:
1519:
1516:Vector product
1513:
1507:
1501:
1487:
1483:
1468:
1457:
1439:
1433:
1427:
1414:
1412:Ricci calculus
1407:
1404:
1403:
1402:
1396:
1390:
1384:
1375:
1369:
1346:
1343:
1342:
1341:
1335:
1329:
1325:
1324:, name for a Z
1319:
1313:
1308:
1302:
1294:
1291:
1290:
1289:
1282:
1279:
1275:charge density
1255:Line integrals
1227:
1224:
1223:
1222:
1215:Logical vector
1212:
1178:
1172:
1166:
1163:Burgers vector
1160:
1111:
1106:
1091:
1090:
1049:
1047:
1040:
1028:
1025:
1024:
1023:
1020:p-adic numbers
1005:
976:, also called
971:
952:
929:
926:
917:Hilbert spaces
851:natural number
824:linear algebra
727:
719:
714:Main article:
711:
708:
699:Pseudo vectors
629:speed of light
623:
581:line of action
501:physical space
452:
444:
439:Main article:
436:
433:
378:distributivity
362:multiplication
306:
299:
294:
291:
273:terminal point
191:
183:
178:Main article:
175:
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2135:Pedoe, Daniel
2132:
2129:
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2110:
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2100:
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2085:1-55369-133-4
2081:
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2017:
2013:
2009:
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1984:
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1951:
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1852:(in Japanese)
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1827:(in Japanese)
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1728:9780195219425
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1697:Jeff Miller.
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1430:Vector bundle
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1345:Vector fields
1344:
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1312:
1311:Hilbert space
1309:
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1203:. However, a
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1179:
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1073:November 2021
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921:Banach spaces
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913:normed spaces
910:
906:
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817:
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794:
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782:
781:vector axioms
778:
774:
773:
768:
764:
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752:
731:
724:
717:
710:Vector spaces
709:
704:
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697:
695:
694:wind velocity
691:
687:
683:
682:
681:vector fields
677:
673:
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654:over time (a
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442:
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424:pseudovectors
421:
417:
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393:
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383:
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375:
374:associativity
371:
370:commutativity
367:
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355:
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343:
339:
335:
330:
326:
323:to the point
322:
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267:
266:initial point
263:
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103:
99:
95:
91:
87:
83:
79:
75:
74:displacements
71:
67:
62:
60:
59:vector spaces
56:
52:
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44:
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19:
2140:
2121:
2094:
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2047:. Retrieved
2045:. 2013-08-20
2042:
2003:
1997:
1986:. Retrieved
1971:
1964:
1953:. Retrieved
1938:
1931:
1911:
1890:. Retrieved
1875:
1854:. Retrieved
1849:
1840:
1829:. Retrieved
1824:
1799:. Retrieved
1795:
1785:
1774:. Retrieved
1770:
1761:
1750:. Retrieved
1746:
1737:
1718:
1713:
1702:. Retrieved
1676:
1669:
1661:
1657:
1648:
1633:
1622:
1611:
1597:Phase vector
1573:Vector meson
1561:Vector boson
1529:
1525:
1445:
1416:
1351:vector field
1348:
1229:
1204:
1199:that may be
1141:
1137:
1094:
1079:
1070:
1051:
985:
978:spin vectors
977:
960:
937:
931:
901:Lie algebras
854:
846:
836:
797:
785:real numbers
780:
771:
766:
759:vector space
748:
723:Vector space
716:Vector space
679:
660:discretizing
641:
620:
616:
593:
580:
574:
567:ordered pair
544:
540:
534:
520:
473:
467:
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412:displacement
400:acceleration
386:vector space
350:real numbers
341:
337:
334:displacement
328:
324:
320:
318:
275:
272:
268:
265:
259:
250:, including
241:
238:vector space
207:
202:
198:
166:
148:
121:
115:, which are
108:
106:
102:real numbers
63:
46:
36:
1616:Ivanov 2001
1399:Vector flow
1239:engineering
1191:, a set of
1008:Witt vector
990:closed loop
955:Multivector
751:mathematics
674:(e.g., the
656:time series
614:component,
563:translation
527:engineering
483:), often a
358:subtraction
218:engineering
210:mathematics
165:. The term
39:mathematics
2061:References
2049:2024-09-08
1988:2024-09-08
1955:2024-09-09
1892:2024-09-08
1856:2024-09-07
1831:2024-09-07
1801:2020-08-19
1776:2020-08-19
1752:2020-08-19
1704:2007-05-25
1693:required.)
1642:Pedoe 1988
1247:divergence
1201:correlated
1189:statistics
1031:See also:
998:orientable
949:quaternion
885:structures
843:isomorphic
806:, such as
668:evaluation
664:trajectory
51:quantities
2145:. Dover.
2128:EMS Press
2099:MIT Press
1767:"Vectors"
1482:∇
1456:∇
1061:talk page
877:have the
871:countably
839:dimension
820:direction
816:magnitude
703:bivectors
600:spacetime
531:mechanics
493:direction
489:magnitude
382:Euclidean
298:⟶
234:direction
226:magnitude
107:The term
90:distances
70:mechanics
2164:Category
2137:(1988).
2118:"Vector"
1792:"Vector"
1668:"vector
1638:ItĂ´ 1993
1544:codomain
1540:function
1406:See also
1359:manifold
1281:See also
1243:gradient
1231:Calculus
1219:Booleans
1195:-valued
1157:rotation
1095:The set
1054:disputed
994:manifold
889:algebras
863:geometry
828:matrices
812:velocity
652:sequence
612:timelike
481:unitless
408:position
396:velocity
366:negation
354:addition
352:such as
159:algebras
82:velocity
66:geometry
1235:physics
1142:vectors
1138:vectors
986:locally
974:Spinors
964:-vector
938:vectors
772:scalars
767:vectors
755:physics
606:, with
585:support
559:newtons
553:on the
523:physics
512:SI unit
474:vector
454:In the
428:tensors
392:physics
344:. Many
271:with a
256:support
214:physics
43:physics
2149:
2105:
2082:
2018:
1979:
1946:
1919:
1883:
1725:
1664:, see
1662:vector
1424:Wilson
1249:, and
1126:tuples
932:Every
808:forces
743:v + 2w
690:region
596:metric
516:meters
472:and a
404:forces
394:: the
376:, and
364:, and
329:vector
232:) and
230:length
216:, and
167:vector
136:axioms
130:and a
113:tuples
109:vector
94:masses
78:forces
55:scalar
47:vector
32:Vector
1687:
1603:Notes
1444:, or
1353:is a
1187:, in
1016:carry
793:field
551:force
510:with
505:three
487:with
336:from
2147:ISBN
2103:ISBN
2080:ISBN
2016:ISBN
1977:ISBN
1944:ISBN
1917:ISBN
1881:ISBN
1723:ISBN
1695:and
1658:veho
1538:, a
1263:flux
1251:curl
1237:and
1193:real
1151:, a
947:, a
919:and
899:and
869:are
810:and
787:and
757:, a
753:and
701:and
686:two-
525:and
491:and
458:, a
426:and
398:and
240:. A
228:(or
220:, a
161:and
98:time
96:and
80:and
41:and
2043:ISO
2008:doi
1528:or
1446:del
1183:or
1128:of
1124:of
957:or
763:set
749:In
579:or
539:or
521:In
514:of
499:in
410:or
348:on
340:to
208:In
201:to
124:set
37:In
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2126:,
2120:,
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2041:.
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624:0
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342:B
338:A
325:B
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293:B
290:A
276:B
269:A
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199:A
189:.
34:.
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