194:
730:
1046:
844:, 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
1244:. 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).
1913:
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,
1236:
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
1125:
1521:, 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
1501:
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
1469:
1143:, 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
2172:
2109:
2022:
1983:
1950:
1923:
1887:
1063:
1429:, 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
2153:
2086:
1729:
1085:
583:, over which the vector quantity can be translated (without rotations). A free vector is a vector quantity having an undefined
1420:
1272:, essential for computations involving scalar or vector fields over three-dimensional regions, contribute to understanding
2130:
594:
Aside from the notion of units and support, physical vector quantities may also differ from
Euclidean vectors in terms of
1187:
1147:
even when addition and scalar multiplication of vectors are not valid operations on these data. Here are some examples.
939:
is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called
881:
2125:
1551:
834:
281:
1435:, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space
1168:, a vector that represents the magnitude and direction of the lattice distortion of dislocation in a crystal lattice
1067:
1056:
833:, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying
1848:
1823:
1680:
1368:
1374:
1340:
907:
841:
786:. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars:
1685:
1389:
1334:
1289:
995:
in the space of rotation vectors may induce a curve in the space of rotations that is not a loop. Also, the
818:
333:
225:
143:
73:
31:
1560:, an econometric model used to capture the evolution and the interdependencies between multiple time series
1542:
1538:
1357:
887:
822:
671:
562:
224:
or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has
139:
1557:
1136:
776:
607:
584:
545:
507:
411:
260:
255:
131:
1590:, a solitary wave with multiple components coupled together that maintains its shape during propagation
868:
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
1101:
573:
and unit (length an meters). A sliding vector is the combination of an ordinary vector quantity and a
1426:
1380:
1318:
936:
891:
830:
770:
689:
603:
535:
492:
251:
233:
158:
1581:
1444:
1384:
1307:
1301:
1159:
1014:
1000:
915:
911:
899:
795:
667:
588:
575:
468:
419:
345:
1940:
1670:
1656:
1479:
1324:
1253:
1177:
1035:
1018:
873:
861:
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
1973:
1877:
1572:, a function defined on a family of sets and taking vector values satisfying certain properties
2149:
2105:
2082:
2018:
1979:
1946:
1919:
1883:
1725:
1524:
1273:
1221:
947:
806:
762:
675:
504:
463:
415:
407:
247:
123:
50:
1693:
1454:
1174:, in musical set theory, an array that expresses the intervallic content of a pitch-class set
2120:
2010:
1593:
1473:
1395:
1261:
1155:
1135:
real numbers has a natural structure of vector space defined by component-wise addition and
1004:
1003:, while the manifold of rotations is not. Spinors are elements of a vector subspace of some
970:
802:
636:
632:
484:
455:
365:
221:
186:
179:
85:
54:
1512:
1506:
1438:
1269:
1211:
1199:
1171:
1151:
984:
895:
869:
791:
685:
647:
643:
595:
554:
496:
475:
459:
447:
440:
381:
242:
154:
150:
127:
116:
1441:, a branch of mathematics concerned with differentiation and integration of vector fields
1745:
2142:
1587:
1569:
1518:
1414:
1277:
1217:
1165:
877:
853:
826:
628:
500:
377:
368:
have close analogues for vectors, operations which obey the familiar algebraic laws of
361:
229:
162:
2014:
2166:
1432:
1313:
1257:
1183:
1022:
919:
698:
693:
373:
369:
2069:
741:(red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum
2137:
1599:
1575:
1563:
1353:
992:
923:
787:
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:
1769:
886:
Many vector spaces that are considered in mathematics are also endowed with other
193:
1401:
1337:, a vector space V equipped with a non-degenerate, skew-symmetric, bilinear form
1241:
1195:
1010:
957:
903:
750:
655:
526:
488:
357:
349:
217:
209:
101:
38:
1343:, a blend of topological structure with the algebraic concept of a vector space
729:
1249:
1203:
1191:
1013:, an infinite sequence of elements of a commutative ring, which belongs to an
951:
845:
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
2076:
17:
2101:
1794:
1404:, 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
2041:
1546:
1361:
1245:
1233:
996:
961:
865:
814:
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
1304:, a type of vector space that includes the extra structure of gradation
1237:
754:
558:
522:
511:
391:
213:
42:
1554:, a linear transformation that converts a matrix into a column vector
1260:, crucial for calculating work along a path within force fields, and
976:
515:
427:
72:) for quantities that have both a magnitude and a direction, such as
1180:, in statistics, a vector with non-negative entries that sum to one.
769:, can be added together and multiplied ("scaled") by numbers called
1701:
1268:, illustrate the practical utility of calculus in vector analysis.
1915:
Mechanical
Systems, Classical Models: Volume 1: Particle Mechanics
1128:
810:
782:
728:
550:
403:
192:
135:
112:
77:
1535:, a linear mapping producing a vector parallel to a second vector
1451:, a vector differential operator represented by the nabla symbol
1371:, a vector field that is the gradient of a scalar potential field
1942:
Analytical
Mechanics: Solutions to Problems in Classical Physics
1265:
97:
93:
2042:"ISO 80000-2:2019 - Quantities and units - Part 2: Mathematics"
1377:, a vector field defined for any energy function or Hamiltonian
1256:, which find applications in physics and engineering contexts.
1039:
2005:"Appendix A. Linear Algebra from a Geometric Point of View".
1978:. Dover Books on Mathematics. Dover Publications. p. 2.
1907:
1905:
1879:
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.
1360:
that, generally, has a domain of the same dimension (as a
149:
Many vector spaces are considered in mathematics, such as
2009:. Ithaca, NY: David W. Henderson. 2013. p. 121–138.
1702:"Earliest Known Uses of Some of the Words of Mathematics"
1515:, a type of differential operator used in vector calculus
610:
of meters: it includes a position
Euclidean vector and a
1972:
Borisenko, A.I.; Tarapov, I.E.; Silverman, R.A. (2012).
1503:, is a differential operator defined over a vector field
2078:
Introduction to Tensor
Calculus and Continuum Mechanics
2036:
2034:
1871:
1869:
794:. Scalars can also be, more generally, elements of any
1818:
1816:
1814:
864:. Finite-dimensional vector spaces occur naturally in
284:
236:. Euclidean vectors can be added and scaled to form a
1824:"Details for IEV number 102-03-21: "vector quantity""
1663:= "I carry". For historical development of the word
1482:
1457:
1104:
1066:. Please help to ensure that disputed statements are
987:. In fact, rotation vectors represent well rotations
1584:, a quantization technique used in signal processing
733:
Vector addition and scalar multiplication: a vector
1398:, a vector field whose curl is a given vector field
983:, have been introduced for extending the notion of
2141:
1669:
1495:
1463:
1119:
825:. 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.
1566:, a boson with the spin quantum number equal to 1
466:. It is typically formulated as the product of a
1724:(2nd. ed.). London: Clarendon Press. 2001.
1509:, 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}}.}
2007: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
1321:, a vector space equipped with a partial order
876:infinite-dimensional vector spaces, and many
8:
1975:Vector and Tensor Analysis with Applications
1849:"Details for IEV number 102-03-04: "vector""
1310:, a vector space on which a norm is defined
1162: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
1629:
1578:, a meson with total spin 1 and odd parity
1476:, 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.
2070:Vectors - The Feynman Lectures on Physics
1918:. Springer Science & Business Media.
1882:. Cambridge University Press. p. 3.
1853:International Electrotechnical Vocabulary
1828:International Electrotechnical Vocabulary
1487:
1481:
1456:
1158:whose direction is that of the axis of a
1111:
1107:
1106:
1103:
1086:Learn how and when to remove this message
840: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
1264:, employed to determine quantities like
1214:that takes its values in a vector space.
1062:Relevant discussion may be found on the
775:. The operations of vector addition and
1611:
1017:, and has been introduced for handling
2098:Encyclopedic Dictionary of Mathematics
1618:
1644:
1392:, a vector field with zero divergence
84:. Such quantities are represented by
7:
1694:participating institution membership
1139:. It is common to call these tuples
943:, mainly due to historical reasons.
666:. A vector may also result from the
111:is also used, in some contexts, for
1912:Teodorescu, Petre P. (2007-06-06).
1640:
1484:
1458:
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:
2173:Vectors (mathematics and physics)
1746:"vector | Definition & Facts"
1425:a textbook on vector calculus by
1296:Vector spaces with more structure
1021:propagation in the operations on
910:, which include function spaces,
2144:Geometry: A comprehensive course
1120:{\displaystyle \mathbb {R} ^{n}}
1044:
721:This section is an excerpt from
446:This section is an excerpt from
185:This section is an excerpt from
2015:10.3792/euclid/9781429799843-13
829:, together with the concept of
1939:Merches, I.; Radu, D. (2014).
991:, but not globally, because a
297:
1:
1722:The Oxford English Dictionary
1545:that has a vector space as a
765:whose elements, often called
174:Vectors in Euclidean geometry
1188:multivariate random variable
882:cardinality of the continuum
658:), such as position vectors
2126:Encyclopedia of Mathematics
1596:, a type of audio synthesis
1552:Vectorization (mathematics)
1496:{\displaystyle \nabla ^{2}}
1030:Data represented by vectors
906:. This is also the case of
835:systems of linear equations
598:. For example, an event in
402:of a moving object and the
2189:
2075:Heinbockel, J. H. (2001).
1945:. CRC Press. p. 379.
1033:
860:, and its dimension is an
805:, 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:
1681:Oxford English Dictionary
1369:Conservative vector field
1220:, a vector of 0s and 1s (
908:topological vector spaces
801:Vector spaces generalize
49:is a term that refers to
27:Element of a vector space
1375:Hamiltonian vector field
1341:Topological vector space
1280:, and fluid flow rates.
602:may be represented as a
2081:. Trafford Publishing.
1750:Encyclopedia Britannica
1686:Oxford University Press
1659:of vehere, "to carry"/
1464:{\displaystyle \nabla }
1390:Solenoidal vector field
1335:Symplectic vector space
1290:Vector (disambiguation)
999:of rotation vectors is
817:, that have not only a
197:A vector pointing from
144:coordinate vector space
2119:Ivanov, A.B. (2001) ,
1539:Vector-valued function
1497:
1465:
1383:, a vector field on a
1358:vector-valued function
1121:
1015:algebra over this ring
890:. This is the case of
852:if its dimension is a
746:
696:over Earth's surface.
672:vector-valued function
311:
205:
140:Euclidean vector space
1799:mathworld.wolfram.com
1558:Vector autoregression
1498:
1466:
1137:scalar multiplication
1122:
954:with a zero real part
848:). 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
2096:ItĂ´, Kiyosi (1993).
1480:
1455:
1381:Killing vector field
1331:-graded vector space
1319:Ordered vector space
1210:may also refer to a
1102:
1055:factual accuracy is
969:, an element of the
937:algebra over a field
912:inner product spaces
900:associative algebras
898:, polynomial rings,
858: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
1793:Weisstein, Eric W.
1684:(Online ed.).
1582:Vector quantization
1445:Vector differential
1385:Riemannian manifold
1364:) as its codomain,
1308:Normed vector space
1302:Graded vector space
1229:Vectors in calculus
856:. Otherwise, it is
807: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
1774:www.mathsisfun.com
1657:perfect participle
1493:
1461:
1325:Super vector space
1178:Probability vector
1117:
1036:Vector (data type)
973:of a vector space.
931:Vectors in algebra
850:finite-dimensional
747:
692:of space, such as
571:quantity dimension
529:, particularly in
307:
258:, formulated as a
206:
134:that satisfy some
2111:978-0-262-59020-4
2024:978-1-4297-9984-3
1985:978-0-486-13190-0
1952:978-1-4822-3940-9
1925:978-1-4020-5442-6
1889:978-0-521-85811-3
1692:(Subscription or
1525:Vector projection
1274:mass distribution
1262:surface integrals
1096:
1095:
1088:
948:Vector quaternion
862:infinite cardinal
803: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
2180:
2159:
2147:
2133:
2115:
2100:(2nd ed.).
2092:
2057:
2056:
2054:
2053:
2038:
2029:
2028:
2002:
1996:
1995:
1993:
1992:
1969:
1963:
1962:
1960:
1959:
1936:
1930:
1929:
1909:
1900:
1899:
1897:
1896:
1876:Rao, A. (2006).
1873:
1864:
1863:
1861:
1860:
1845:
1839:
1838:
1836:
1835:
1820:
1809:
1808:
1806:
1805:
1790:
1784:
1783:
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1766:
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1759:
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1756:
1742:
1736:
1735:
1718:
1712:
1711:
1709:
1708:
1697:
1689:
1677:
1653:
1647:
1643:, p. 1678;
1638:
1632:
1627:
1621:
1616:
1594:Vector synthesis
1533:vector component
1527:, also known as
1502:
1500:
1499:
1494:
1492:
1491:
1474:Vector Laplacian
1470:
1468:
1467:
1462:
1396:Vector potential
1270:Volume integrals
1200:random variables
1156:Euclidean vector
1134:
1126:
1124:
1123:
1118:
1116:
1115:
1110:
1091:
1084:
1080:
1077:
1071:
1068:reliably sourced
1048:
1047:
1040:
1005:Clifford algebra
971:exterior algebra
966:
896:field extensions
894:, which include
884:as a dimension.
870: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:
2188:
2187:
2183:
2182:
2181:
2179:
2178:
2177:
2163:
2162:
2156:
2136:
2118:
2112:
2095:
2089:
2074:
2066:
2061:
2060:
2051:
2049:
2040:
2039:
2032:
2025:
2004:
2003:
1999:
1990:
1988:
1986:
1971:
1970:
1966:
1957:
1955:
1953:
1938:
1937:
1933:
1926:
1911:
1910:
1903:
1894:
1892:
1890:
1875:
1874:
1867:
1858:
1856:
1847:
1846:
1842:
1833:
1831:
1822:
1821:
1812:
1803:
1801:
1792:
1791:
1787:
1778:
1776:
1768:
1767:
1763:
1754:
1752:
1744:
1743:
1739:
1732:
1720:
1719:
1715:
1706:
1704:
1699:
1691:
1668:
1655:Latin: vectus,
1654:
1650:
1639:
1635:
1630:Heinbockel 2001
1628:
1624:
1617:
1613:
1608:
1529:vector resolute
1513:Vector operator
1507:Vector notation
1483:
1478:
1477:
1453:
1452:
1439:Vector calculus
1421:Vector Analysis
1411:
1350:
1330:
1298:
1286:
1231:
1212:random variable
1172:Interval vector
1152:Rotation vector
1132:
1105:
1100:
1099:
1092:
1081:
1075:
1072:
1061:
1053:This section's
1049:
1045:
1038:
1032:
985:rotation vector
962:
933:
928:
927:
878:function spaces
792: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:
2186:
2184:
2176:
2175:
2165:
2164:
2161:
2160:
2154:
2134:
2116:
2110:
2093:
2087:
2072:
2065:
2062:
2059:
2058:
2030:
2023:
1997:
1984:
1964:
1951:
1931:
1924:
1901:
1888:
1865:
1840:
1810:
1785:
1761:
1737:
1730:
1713:
1648:
1633:
1622:
1610:
1609:
1607:
1604:
1603:
1602:
1597:
1591:
1588:Vector soliton
1585:
1579:
1573:
1570:Vector measure
1567:
1561:
1555:
1549:
1536:
1522:
1519:Vector product
1516:
1510:
1504:
1490:
1486:
1471:
1460:
1442:
1436:
1430:
1417:
1415:Ricci calculus
1410:
1407:
1406:
1405:
1399:
1393:
1387:
1378:
1372:
1349:
1346:
1345:
1344:
1338:
1332:
1328:
1327:, name for a Z
1322:
1316:
1311:
1305:
1297:
1294:
1293:
1292:
1285:
1282:
1278:charge density
1258:Line integrals
1230:
1227:
1226:
1225:
1218:Logical vector
1215:
1181:
1175:
1169:
1166:Burgers vector
1163:
1114:
1109:
1094:
1093:
1052:
1050:
1043:
1031:
1028:
1027:
1026:
1023:p-adic numbers
1008:
979:, also called
974:
955:
932:
929:
920:Hilbert spaces
854:natural number
827: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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2155:0-486-65812-0
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2138:Pedoe, Daniel
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2132:
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2122:
2117:
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2088:1-55369-133-4
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2020:
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2012:
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1855:(in Japanese)
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1830:(in Japanese)
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1731:9780195219425
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1703:
1700:Jeff Miller.
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1433:Vector bundle
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1348:Vector fields
1347:
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1314:Hilbert space
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1182:
1179:
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1076:November 2021
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924:Banach spaces
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710:Vector spaces
709:
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695:
694:wind velocity
691:
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681:vector fields
677:
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654:over time (a
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424:pseudovectors
421:
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374:associativity
371:
370:commutativity
367:
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335:
330:
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323:to the point
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266:initial point
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74:displacements
71:
67:
62:
60:
59:vector spaces
56:
52:
48:
44:
40:
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19:
18:Vector (math)
2143:
2124:
2097:
2077:
2050:. Retrieved
2048:. 2013-08-20
2045:
2006:
2000:
1989:. Retrieved
1974:
1967:
1956:. Retrieved
1941:
1934:
1914:
1893:. Retrieved
1878:
1857:. Retrieved
1852:
1843:
1832:. Retrieved
1827:
1802:. Retrieved
1798:
1788:
1777:. Retrieved
1773:
1764:
1753:. Retrieved
1749:
1740:
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1716:
1705:. Retrieved
1679:
1672:
1664:
1660:
1651:
1636:
1625:
1614:
1600:Phase vector
1576:Vector meson
1564:Vector boson
1532:
1528:
1448:
1419:
1354:vector field
1351:
1232:
1207:
1202:that may be
1144:
1140:
1097:
1082:
1073:
1054:
988:
981:spin vectors
980:
963:
940:
934:
904:Lie algebras
857:
849:
839:
800:
788: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:
1619:Ivanov 2001
1402:Vector flow
1242:engineering
1194:, a set of
1011:Witt vector
993:closed loop
958: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
2064:References
2052:2024-09-08
1991:2024-09-08
1958:2024-09-09
1895:2024-09-08
1859:2024-09-07
1834:2024-09-07
1804:2020-08-19
1779:2020-08-19
1755:2020-08-19
1707:2007-05-25
1696:required.)
1645:Pedoe 1988
1250:divergence
1204:correlated
1192:statistics
1034:See also:
1001:orientable
952:quaternion
888:structures
846:isomorphic
809:, such as
668:evaluation
664:trajectory
51:quantities
2148:. Dover.
2131:EMS Press
2102:MIT Press
1770:"Vectors"
1485:∇
1459:∇
1064:talk page
880:have the
874:countably
842:dimension
823:direction
819: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
2167:Category
2140:(1988).
2121:"Vector"
1795:"Vector"
1671:"vector
1641:ItĂ´ 1993
1547:codomain
1543:function
1409:See also
1362:manifold
1284:See also
1246:gradient
1234:Calculus
1222:Booleans
1198:-valued
1160:rotation
1098:The set
1057:disputed
997:manifold
892:algebras
866:geometry
831:matrices
815:velocity
652:sequence
612:timelike
481:unitless
408:position
396:velocity
366:negation
354:addition
352:such as
159:algebras
82:velocity
66:geometry
1238:physics
1145:vectors
1141:vectors
989:locally
977:Spinors
967:-vector
941:vectors
781:vector
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
2152:
2108:
2085:
2021:
1982:
1949:
1922:
1886:
1728:
1667:, see
1665:vector
1427:Wilson
1252:, and
1129:tuples
935:Every
811:forces
783:axioms
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
1690:
1606:Notes
1447:, or
1356:is a
1190:, in
1019:carry
796:field
551:force
510:with
505:three
487:with
336:from
2150:ISBN
2106:ISBN
2083:ISBN
2019:ISBN
1980:ISBN
1947:ISBN
1920:ISBN
1884:ISBN
1726:ISBN
1698:and
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1541:, a
1266:flux
1254:curl
1240:and
1196:real
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950:, a
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902:and
872:are
813:and
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686:two-
525:and
491:and
458:, a
426:and
398:and
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228:(or
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161:and
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96:and
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2011:doi
1531:or
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960:or
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579:or
539:or
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410:or
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