1356:
1123:
4318:
3922:
517:
471:
1351:{\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} =\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}
4582:
3565:
2948:
3195:
2621:
47:
2783:
3917:{\displaystyle P_{\mathbf {a} }=\mathbf {a} \mathbf {a} ^{\textsf {T}}={\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}}{\begin{bmatrix}a_{x}&a_{y}&a_{z}\end{bmatrix}}={\begin{bmatrix}a_{x}^{2}&a_{x}a_{y}&a_{x}a_{z}\\a_{x}a_{y}&a_{y}^{2}&a_{y}a_{z}\\a_{x}a_{z}&a_{y}a_{z}&a_{z}^{2}\\\end{bmatrix}}}
2416:
2498:
3993:
For a given vector and plane, the sum of projection and rejection is equal to the original vector. Similarly, for inner product spaces with more than three dimensions, the notions of projection onto a vector and rejection from a vector can be generalized to the notions of projection onto a
3185:
3096:
2178:
466:
2283:
2307:
1697:
1974:
2081:
2711:
2943:{\displaystyle a_{2}=\left\|\mathbf {a} \right\|\sin \theta ={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} _{y}\mathbf {b} _{x}-\mathbf {a} _{x}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}
2493:
1019:
2778:
1817:
3976:
In some cases, the inner product coincides with the dot product. Whenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. For a three-dimensional
652:
597:
3105:
3017:
2093:
3989:
on that plane. The rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. Both are vectors. The first is parallel to the plane, the second is orthogonal.
400:
2185:
2616:{\displaystyle \mathbf {a} _{1}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}
377:
267:
340:
182:
815:
3010:
1888:
1539:
1605:
1895:
883:
2018:
2421:
1424:
949:
2411:{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}},}
1753:
2639:
735:
681:
1768:
760:
706:
1118:
1096:
2716:
1724:
910:
3981:, the notions of projection of a vector onto another and rejection of a vector from another can be generalized to the notions of projection of a vector onto a
602:
550:
3973:, this is also true for the notions of orthogonal projection of a vector, projection of a vector onto another, and rejection of a vector from another.
64:
56:
4176:
2963:
1841:
1489:
1063:. The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is
4567:
4104:
3180:{\displaystyle \mathbf {a} _{2}={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\mathbf {b} \cdot \mathbf {b} }}\mathbf {b} ^{\perp }}
345:
235:
4611:
308:
150:
3091:{\displaystyle \mathbf {a} _{2}=\mathbf {a} -{\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }.}
765:
4059:
2173:{\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}.}
4557:
461:{\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .}
4616:
4519:
4455:
2278:{\displaystyle a_{1}={\frac {\mathbf {a} _{x}\mathbf {b} _{x}+\mathbf {a} _{y}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}
834:
4606:
4297:
4169:
4402:
4252:
3537:
1692:{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} =(\left\|\mathbf {a} \right\|\cos \theta )\mathbf {\hat {b}} }
31:
1969:{\displaystyle \mathbf {a} _{2}=\mathbf {a} -\left(\left\|\mathbf {a} \right\|\cos \theta \right)\mathbf {\hat {b}} }
4149:
4133:
1399:
4307:
4201:
4547:
4196:
2076:{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta }
4422:
3932:
4539:
35:
4585:
4292:
4162:
3946:
4349:
4282:
4272:
3942:
2706:{\displaystyle \mathbf {b} ^{\perp }={\begin{pmatrix}-\mathbf {b} _{y}&\mathbf {b} _{x}\end{pmatrix}}}
1729:
4509:
4364:
4359:
4354:
4287:
4232:
3986:
116:
2488:{\displaystyle \mathbf {a} _{1}=\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ,}
711:
657:
4374:
4339:
4326:
4217:
2090:
By the above-mentioned property of the dot product, the definition of the scalar projection becomes:
4552:
4432:
4407:
4257:
4080:
3978:
3970:
1014:{\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} }
740:
686:
1101:
1079:
4262:
2773:{\displaystyle \mathbf {b} ={\begin{pmatrix}\mathbf {b} _{x}&\mathbf {b} _{y}\end{pmatrix}}}
4460:
4417:
4344:
4055:
4023:
4007:
4003:
3232:
3203:
1469:
914:
101:
4049:
4465:
4369:
4222:
3982:
516:
288:
1702:
888:
4524:
4317:
4277:
4267:
4028:
1064:
470:
4529:
4514:
4450:
4185:
3985:, and rejection of a vector from a plane. The projection of a vector on a plane is its
3935:
3262:
3258:
3219:
1812:{\displaystyle \mathbf {\hat {b}} ={\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}}
1034:
4600:
4562:
4485:
4445:
4412:
4392:
205:
126:
647:{\displaystyle \mathbf {a} _{2}:=\operatorname {oproj} _{\mathbf {b} }\mathbf {a} .}
4495:
4384:
4334:
4227:
3939:
1561:
592:{\displaystyle \mathbf {a} _{1}:=\operatorname {proj} _{\mathbf {b} }\mathbf {a} }
4475:
4440:
4397:
4242:
3250:
2004:
1756:
1026:
937:
1365:
This article uses the convention that vectors are denoted in a bold font (e.g.
17:
4504:
4247:
3999:
3995:
296:
292:
4302:
2630:
In two dimensions, the scalar rejection is equivalent to the projection of
831:
can be decomposed into a direction and a scalar magnitude by writing it as
4470:
3272:
of the vector projection if the angle is smaller than 90°. More exactly:
3194:
4480:
3958:
1073:, that is, if the angle between the vectors is more than 90 degrees.
3100:
By using the Scalar rejection using the perp dot product this gives
3962:
3193:
1048:
4154:
1076:
The vector projection can be calculated using the dot product of
372:{\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} }
262:{\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} }
335:{\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} }
177:{\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} }
4158:
810:{\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}.}
3562:, it would need to be multiplied with this projection matrix:
3005:{\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}
1883:{\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}
1726:
is the corresponding scalar projection, as defined above, and
1534:{\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta ,}
40:
878:{\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} }
1584:
is a vector whose magnitude is the scalar projection of
3931:
The vector projection is an important operation in the
68:
3718:
3665:
3611:
2733:
2663:
2292:
Similarly, the definition of the vector projection of
3568:
3108:
3020:
2966:
2952:
Such a dot product is called the "perp dot product."
2786:
2719:
2642:
2501:
2424:
2310:
2188:
2096:
2021:
1898:
1844:
1771:
1732:
1705:
1608:
1492:
1402:
1126:
1104:
1082:
952:
891:
837:
768:
743:
714:
689:
660:
605:
553:
403:
348:
311:
238:
153:
4006:, they can be further generalized to the notions of
4538:
4494:
4431:
4383:
4325:
4210:
4051:Geometric Algebra With Applications in Engineering
4010:of a general multivector onto/from any invertible
3916:
3536:The orthogonal projection can be represented by a
3179:
3090:
3004:
2942:
2772:
2705:
2615:
2487:
2410:
2277:
2172:
2075:
1968:
1882:
1811:
1747:
1718:
1691:
1533:
1418:
1350:
1112:
1090:
1013:
904:
877:
809:
754:
729:
700:
675:
646:
591:
460:
371:
334:
261:
176:
2476:
2459:
2342:
1960:
1778:
1739:
1683:
1640:
1374:), and scalars are written in normal font (e.g.
1186:
1169:
1005:
869:
1564:to compute the corresponding vector projection.
3949:to detect whether two convex shapes intersect.
4122:. San Diego: Academic Press. pp. 138–148.
1419:{\displaystyle \mathbf {a} \cdot \mathbf {b} }
4170:
8:
3540:. To project a vector onto the unit vector
547:To simplify notation, this article defines
199:The vector component or vector resolute of
4177:
4163:
4155:
3965:between vectors can be generalized to any
537:has an opposite direction with respect to
3900:
3895:
3883:
3873:
3861:
3851:
3837:
3827:
3815:
3810:
3798:
3788:
3774:
3764:
3752:
3742:
3730:
3725:
3713:
3696:
3684:
3672:
3660:
3646:
3632:
3618:
3606:
3597:
3596:
3595:
3590:
3584:
3574:
3573:
3567:
3249:is a scalar which has a negative sign if
3171:
3166:
3157:
3149:
3141:
3136:
3127:
3124:
3115:
3110:
3107:
3079:
3078:
3070:
3062:
3055:
3047:
3044:
3036:
3027:
3022:
3019:
2996:
2991:
2982:
2973:
2968:
2965:
2926:
2915:
2910:
2903:
2898:
2888:
2883:
2876:
2871:
2867:
2853:
2842:
2837:
2828:
2825:
2804:
2791:
2785:
2756:
2751:
2742:
2737:
2728:
2720:
2718:
2689:
2684:
2675:
2670:
2658:
2649:
2644:
2641:
2601:
2600:
2592:
2584:
2577:
2569:
2566:
2557:
2556:
2548:
2539:
2528:
2520:
2517:
2508:
2503:
2500:
2471:
2470:
2454:
2453:
2445:
2431:
2426:
2423:
2394:
2385:
2383:
2372:
2362:
2354:
2351:
2337:
2336:
2330:
2317:
2312:
2309:
2261:
2250:
2245:
2238:
2233:
2223:
2218:
2211:
2206:
2202:
2193:
2187:
2156:
2146:
2138:
2135:
2114:
2101:
2095:
2055:
2042:
2030:
2022:
2020:
1955:
1954:
1931:
1914:
1905:
1900:
1897:
1874:
1869:
1860:
1851:
1846:
1843:
1798:
1789:
1787:
1773:
1772:
1770:
1734:
1733:
1731:
1710:
1704:
1678:
1677:
1656:
1635:
1634:
1628:
1615:
1610:
1607:
1510:
1497:
1491:
1411:
1403:
1401:
1336:
1335:
1327:
1319:
1312:
1304:
1301:
1292:
1291:
1283:
1274:
1263:
1255:
1252:
1238:
1229:
1227:
1216:
1206:
1198:
1195:
1181:
1180:
1164:
1163:
1155:
1142:
1132:
1131:
1125:
1105:
1103:
1083:
1081:
1000:
999:
991:
970:
957:
951:
896:
890:
864:
863:
857:
844:
839:
836:
798:
793:
783:
778:
769:
767:
744:
742:
721:
716:
713:
690:
688:
667:
662:
659:
636:
626:
625:
612:
607:
604:
584:
574:
573:
560:
555:
552:
450:
440:
439:
427:
419:
409:
408:
402:
364:
354:
353:
347:
327:
317:
316:
310:
254:
244:
243:
237:
169:
159:
158:
152:
515:
469:
4040:
1826:By definition, the vector rejection of
379:are vectors, and their sum is equal to
4568:Comparison of linear algebra libraries
3466:which is either null or orthogonal to
946:. The scalar projection is defined as
1560:A scalar projection can be used as a
7:
4136:Published on www.euclideanspace.com.
4134:Projection of a vector onto a plane.
4075:
4073:
4071:
3357:which is either null or parallel to
69:move details into the article's body
4150:Projection of a vector onto a plane
2003:, by the following property of the
1748:{\displaystyle \mathbf {\hat {b}} }
281:), is the orthogonal projection of
25:
4581:
4580:
4558:Basic Linear Algebra Subprograms
4316:
3591:
3585:
3575:
3167:
3158:
3150:
3137:
3128:
3111:
3080:
3071:
3063:
3056:
3048:
3037:
3023:
2992:
2983:
2969:
2927:
2911:
2899:
2884:
2872:
2854:
2838:
2829:
2805:
2780:rotated 90° to the left. Hence,
2752:
2738:
2721:
2685:
2671:
2645:
2602:
2593:
2585:
2578:
2570:
2558:
2540:
2529:
2521:
2504:
2473:
2456:
2446:
2427:
2395:
2386:
2373:
2363:
2355:
2339:
2313:
2262:
2246:
2234:
2219:
2207:
2182:In two dimensions, this becomes
2157:
2147:
2139:
2115:
2056:
2043:
2031:
2023:
1957:
1932:
1915:
1901:
1870:
1861:
1847:
1799:
1790:
1775:
1736:
1680:
1657:
1637:
1611:
1511:
1412:
1404:
1337:
1328:
1320:
1313:
1305:
1293:
1275:
1264:
1256:
1239:
1230:
1217:
1207:
1199:
1183:
1166:
1156:
1143:
1133:
1106:
1084:
1002:
992:
971:
866:
840:
794:
779:
770:
745:
730:{\displaystyle \mathbf {a} _{2}}
717:
691:
676:{\displaystyle \mathbf {a} _{1}}
663:
637:
627:
608:
585:
575:
556:
451:
441:
428:
420:
410:
365:
355:
328:
318:
255:
245:
170:
160:
45:
4456:Seven-dimensional cross product
4081:"Scalar and Vector Projections"
1979:Definitions in terms of a and b
30:For more general concepts, see
4105:"Dot Products and Projections"
2931:
2923:
2858:
2850:
2809:
2801:
2544:
2536:
2418:which is equivalent to either
2399:
2391:
2377:
2369:
2266:
2258:
2161:
2153:
2119:
2111:
2060:
2052:
2047:
2039:
1936:
1928:
1803:
1795:
1674:
1661:
1653:
1649:
1515:
1507:
1279:
1271:
1243:
1235:
1221:
1213:
975:
967:
109:on (or onto) a nonzero vector
1:
755:{\displaystyle \mathbf {b} ,}
701:{\displaystyle \mathbf {b} ,}
4298:Eigenvalues and eigenvectors
3957:Since the notions of vector
3428:have opposite directions if
3202:≤ 90°, as in this case, the
1991:can be computed in terms of
1987:is not known, the cosine of
1113:{\displaystyle \mathbf {b} }
1091:{\displaystyle \mathbf {a} }
214:, sometimes also called the
3403:have the same direction if
1759:with the same direction as
1602:. Namely, it is defined as
1596:with the same direction as
1456:Definitions based on angle
1384:The dot product of vectors
32:Projection (linear algebra)
4633:
3230:
1467:
29:
4612:Transformation (function)
4576:
4314:
4192:
3945:. It is also used in the
3336:The vector projection of
3222:of the vector projection.
1572:The vector projection of
1474:The scalar projection of
27:Concept in linear algebra
4118:Hill, F. S. Jr. (1994).
4008:projection and rejection
3445:The vector rejection of
3261:. It coincides with the
912:is a scalar, called the
36:Projection (mathematics)
3998:, and rejection from a
3947:separating axis theorem
4617:Functions and mappings
4283:Row and column vectors
3918:
3237:The scalar projection
3223:
3181:
3092:
3006:
2944:
2774:
2707:
2617:
2489:
2412:
2279:
2174:
2077:
1970:
1884:
1813:
1749:
1720:
1693:
1535:
1420:
1352:
1114:
1092:
1015:
906:
879:
811:
756:
731:
702:
677:
648:
593:
544:
513:
462:
373:
336:
263:
178:
4607:Operations on vectors
4288:Row and column spaces
4233:Scalar multiplication
3987:orthogonal projection
3919:
3532:Matrix representation
3197:
3182:
3093:
3007:
2945:
2775:
2708:
2618:
2490:
2413:
2280:
2175:
2078:
1971:
1885:
1814:
1750:
1721:
1719:{\displaystyle a_{1}}
1694:
1545:is the angle between
1536:
1486:is a scalar equal to
1436:‖, the angle between
1421:
1353:
1115:
1093:
1016:
907:
905:{\displaystyle a_{1}}
880:
812:
757:
732:
703:
678:
649:
594:
519:
473:
463:
374:
337:
264:
179:
135:. The projection of
117:orthogonal projection
4423:Gram–Schmidt process
4375:Gaussian elimination
4048:Perwass, G. (2009).
3566:
3106:
3018:
2964:
2784:
2717:
2640:
2499:
2422:
2308:
2186:
2094:
2019:
1896:
1842:
1769:
1730:
1703:
1606:
1490:
1400:
1124:
1102:
1080:
1067:to the direction of
950:
940:in the direction of
889:
835:
766:
741:
712:
687:
658:
603:
551:
493:), and rejection of
401:
346:
309:
236:
151:
147:is often written as
4553:Numerical stability
4433:Multilinear algebra
4408:Inner product space
4258:Linear independence
3979:inner product space
3971:inner product space
3905:
3820:
3735:
3218:coincides with the
1021:where the operator
385:, the rejection of
92:(also known as the
4263:Linear combination
4132:M.J. Baker, 2012.
3936:orthonormalization
3914:
3908:
3891:
3806:
3721:
3704:
3654:
3224:
3177:
3088:
3002:
2940:
2770:
2764:
2703:
2697:
2613:
2485:
2408:
2275:
2170:
2073:
1966:
1880:
1809:
1745:
1716:
1689:
1531:
1416:
1348:
1110:
1088:
1011:
902:
875:
819:The projection of
807:
752:
727:
698:
673:
644:
589:
545:
514:
458:
369:
332:
259:
174:
4594:
4593:
4461:Geometric algebra
4418:Kronecker product
4253:Linear projection
4238:Vector projection
4024:Scalar projection
4004:geometric algebra
3599:
3538:projection matrix
3513:is orthogonal to
3332:Vector projection
3233:Scalar projection
3227:Scalar projection
3204:scalar projection
3163:
3076:
2935:
2862:
2609:
2598:
2554:
2479:
2462:
2403:
2381:
2345:
2288:Vector projection
2270:
2165:
2086:Scalar projection
1963:
1807:
1781:
1742:
1686:
1643:
1568:Vector projection
1470:Scalar projection
1464:Scalar projection
1344:
1333:
1289:
1247:
1225:
1189:
1172:
1008:
915:scalar projection
872:
737:is orthogonal to
654:Thus, the vector
291:(or, in general,
98:vector resolution
90:vector projection
86:
85:
65:length guidelines
16:(Redirected from
4624:
4584:
4583:
4466:Exterior algebra
4403:Hadamard product
4320:
4308:Linear equations
4179:
4172:
4165:
4156:
4137:
4130:
4124:
4123:
4120:Graphics Gems IV
4115:
4109:
4108:
4101:
4095:
4094:
4092:
4091:
4077:
4066:
4065:
4045:
3923:
3921:
3920:
3915:
3913:
3912:
3904:
3899:
3888:
3887:
3878:
3877:
3866:
3865:
3856:
3855:
3842:
3841:
3832:
3831:
3819:
3814:
3803:
3802:
3793:
3792:
3779:
3778:
3769:
3768:
3757:
3756:
3747:
3746:
3734:
3729:
3709:
3708:
3701:
3700:
3689:
3688:
3677:
3676:
3659:
3658:
3651:
3650:
3637:
3636:
3623:
3622:
3602:
3601:
3600:
3594:
3588:
3580:
3579:
3578:
3561:
3526:
3518:
3512:
3501:
3494:
3487:
3472:. More exactly:
3471:
3465:
3456:
3450:
3441:Vector rejection
3435:
3427:
3421:
3410:
3402:
3396:
3385:
3378:
3363:. More exactly:
3362:
3356:
3347:
3341:
3326:
3318:
3299:
3291:
3271:
3259:180 degrees
3248:
3242:
3217:
3211:
3186:
3184:
3183:
3178:
3176:
3175:
3170:
3164:
3162:
3161:
3153:
3147:
3146:
3145:
3140:
3131:
3125:
3120:
3119:
3114:
3097:
3095:
3094:
3089:
3084:
3083:
3077:
3075:
3074:
3066:
3060:
3059:
3051:
3045:
3040:
3032:
3031:
3026:
3011:
3009:
3008:
3003:
3001:
3000:
2995:
2986:
2978:
2977:
2972:
2956:Vector rejection
2949:
2947:
2946:
2941:
2936:
2934:
2930:
2921:
2920:
2919:
2914:
2908:
2907:
2902:
2893:
2892:
2887:
2881:
2880:
2875:
2868:
2863:
2861:
2857:
2848:
2847:
2846:
2841:
2832:
2826:
2812:
2808:
2796:
2795:
2779:
2777:
2776:
2771:
2769:
2768:
2761:
2760:
2755:
2747:
2746:
2741:
2724:
2712:
2710:
2709:
2704:
2702:
2701:
2694:
2693:
2688:
2680:
2679:
2674:
2654:
2653:
2648:
2635:
2626:Scalar rejection
2622:
2620:
2619:
2614:
2607:
2606:
2605:
2599:
2597:
2596:
2588:
2582:
2581:
2573:
2567:
2562:
2561:
2555:
2553:
2552:
2547:
2543:
2533:
2532:
2524:
2518:
2513:
2512:
2507:
2494:
2492:
2491:
2486:
2481:
2480:
2472:
2469:
2465:
2464:
2463:
2455:
2449:
2436:
2435:
2430:
2417:
2415:
2414:
2409:
2404:
2402:
2398:
2389:
2384:
2382:
2380:
2376:
2367:
2366:
2358:
2352:
2347:
2346:
2338:
2335:
2334:
2322:
2321:
2316:
2303:
2297:
2284:
2282:
2281:
2276:
2271:
2269:
2265:
2256:
2255:
2254:
2249:
2243:
2242:
2237:
2228:
2227:
2222:
2216:
2215:
2210:
2203:
2198:
2197:
2179:
2177:
2176:
2171:
2166:
2164:
2160:
2151:
2150:
2142:
2136:
2122:
2118:
2106:
2105:
2082:
2080:
2079:
2074:
2063:
2059:
2050:
2046:
2034:
2026:
2015:
2002:
1996:
1990:
1986:
1975:
1973:
1972:
1967:
1965:
1964:
1956:
1953:
1949:
1939:
1935:
1918:
1910:
1909:
1904:
1889:
1887:
1886:
1881:
1879:
1878:
1873:
1864:
1856:
1855:
1850:
1837:
1831:
1822:Vector rejection
1818:
1816:
1815:
1810:
1808:
1806:
1802:
1793:
1788:
1783:
1782:
1774:
1764:
1754:
1752:
1751:
1746:
1744:
1743:
1735:
1725:
1723:
1722:
1717:
1715:
1714:
1698:
1696:
1695:
1690:
1688:
1687:
1679:
1664:
1660:
1645:
1644:
1636:
1633:
1632:
1620:
1619:
1614:
1601:
1595:
1589:
1583:
1577:
1556:
1550:
1540:
1538:
1537:
1532:
1518:
1514:
1502:
1501:
1485:
1479:
1447:
1441:
1431:
1425:
1423:
1422:
1417:
1415:
1407:
1395:
1389:
1373:
1357:
1355:
1354:
1349:
1342:
1341:
1340:
1334:
1332:
1331:
1323:
1317:
1316:
1308:
1302:
1297:
1296:
1290:
1288:
1287:
1282:
1278:
1268:
1267:
1259:
1253:
1248:
1246:
1242:
1233:
1228:
1226:
1224:
1220:
1211:
1210:
1202:
1196:
1191:
1190:
1182:
1179:
1175:
1174:
1173:
1165:
1159:
1146:
1138:
1137:
1136:
1119:
1117:
1116:
1111:
1109:
1097:
1095:
1094:
1089:
1087:
1072:
1062:
1056:
1042:
1020:
1018:
1017:
1012:
1010:
1009:
1001:
995:
978:
974:
962:
961:
945:
935:
929:
923:
911:
909:
908:
903:
901:
900:
884:
882:
881:
876:
874:
873:
865:
862:
861:
849:
848:
843:
830:
824:
816:
814:
813:
808:
803:
802:
797:
788:
787:
782:
773:
761:
759:
758:
753:
748:
736:
734:
733:
728:
726:
725:
720:
707:
705:
704:
699:
694:
682:
680:
679:
674:
672:
671:
666:
653:
651:
650:
645:
640:
632:
631:
630:
617:
616:
611:
598:
596:
595:
590:
588:
580:
579:
578:
565:
564:
559:
542:
536:
527:
504:
498:
485:
479:
467:
465:
464:
459:
454:
446:
445:
444:
431:
423:
415:
414:
413:
396:
390:
384:
378:
376:
375:
370:
368:
360:
359:
358:
341:
339:
338:
333:
331:
323:
322:
321:
304:
286:
280:
268:
266:
265:
260:
258:
250:
249:
248:
231:
223:
216:vector rejection
213:
204:
195:
183:
181:
180:
175:
173:
165:
164:
163:
146:
140:
134:
124:
114:
108:
94:vector component
81:
78:
72:
63:Please read the
49:
48:
41:
21:
4632:
4631:
4627:
4626:
4625:
4623:
4622:
4621:
4597:
4596:
4595:
4590:
4572:
4534:
4490:
4427:
4379:
4321:
4312:
4278:Change of basis
4268:Multilinear map
4206:
4188:
4183:
4146:
4141:
4140:
4131:
4127:
4117:
4116:
4112:
4103:
4102:
4098:
4089:
4087:
4079:
4078:
4069:
4062:
4047:
4046:
4042:
4037:
4029:Vector notation
4020:
3955:
3953:Generalizations
3929:
3907:
3906:
3889:
3879:
3869:
3867:
3857:
3847:
3844:
3843:
3833:
3823:
3821:
3804:
3794:
3784:
3781:
3780:
3770:
3760:
3758:
3748:
3738:
3736:
3714:
3703:
3702:
3692:
3690:
3680:
3678:
3668:
3661:
3653:
3652:
3642:
3639:
3638:
3628:
3625:
3624:
3614:
3607:
3589:
3569:
3564:
3563:
3558:
3554:
3550:
3541:
3534:
3520:
3514:
3511:
3505:
3496:
3489:
3482:
3476:
3467:
3464:
3458:
3452:
3446:
3443:
3429:
3423:
3420:
3414:
3404:
3398:
3395:
3389:
3380:
3373:
3367:
3358:
3355:
3349:
3343:
3337:
3334:
3320:
3316:
3309:
3303:
3293:
3289:
3282:
3276:
3265:
3251:90 degrees
3244:
3238:
3235:
3229:
3213:
3207:
3192:
3165:
3148:
3135:
3126:
3109:
3104:
3103:
3061:
3046:
3021:
3016:
3015:
2990:
2967:
2962:
2961:
2960:By definition,
2958:
2922:
2909:
2897:
2882:
2870:
2869:
2849:
2836:
2827:
2800:
2787:
2782:
2781:
2763:
2762:
2750:
2748:
2736:
2729:
2715:
2714:
2696:
2695:
2683:
2681:
2669:
2659:
2643:
2638:
2637:
2631:
2628:
2583:
2568:
2535:
2534:
2519:
2502:
2497:
2496:
2444:
2440:
2425:
2420:
2419:
2390:
2368:
2353:
2326:
2311:
2306:
2305:
2299:
2293:
2290:
2257:
2244:
2232:
2217:
2205:
2204:
2189:
2184:
2183:
2152:
2137:
2110:
2097:
2092:
2091:
2088:
2051:
2038:
2017:
2016:
2007:
1998:
1992:
1988:
1984:
1981:
1927:
1926:
1922:
1899:
1894:
1893:
1868:
1845:
1840:
1839:
1833:
1827:
1824:
1794:
1767:
1766:
1760:
1728:
1727:
1706:
1701:
1700:
1652:
1624:
1609:
1604:
1603:
1597:
1591:
1585:
1579:
1573:
1570:
1552:
1546:
1506:
1493:
1488:
1487:
1481:
1475:
1472:
1466:
1461:
1443:
1437:
1427:
1398:
1397:
1391:
1385:
1380:
1372:
1366:
1363:
1318:
1303:
1270:
1269:
1254:
1234:
1212:
1197:
1154:
1150:
1127:
1122:
1121:
1100:
1099:
1078:
1077:
1068:
1058:
1052:
1038:
966:
953:
948:
947:
941:
931:
925:
919:
892:
887:
886:
853:
838:
833:
832:
826:
820:
792:
777:
764:
763:
739:
738:
715:
710:
709:
685:
684:
683:is parallel to
661:
656:
655:
621:
606:
601:
600:
569:
554:
549:
548:
538:
535:
529:
521:
511:
500:
494:
492:
481:
475:
435:
404:
399:
398:
392:
386:
380:
349:
344:
343:
312:
307:
306:
300:
282:
279:
270:
239:
234:
233:
227:
219:
209:
200:
194:
185:
154:
149:
148:
142:
136:
130:
120:
110:
104:
82:
76:
73:
62:
59:may be too long
54:This article's
50:
46:
39:
28:
23:
22:
18:Scalar resolute
15:
12:
11:
5:
4630:
4628:
4620:
4619:
4614:
4609:
4599:
4598:
4592:
4591:
4589:
4588:
4577:
4574:
4573:
4571:
4570:
4565:
4560:
4555:
4550:
4548:Floating-point
4544:
4542:
4536:
4535:
4533:
4532:
4530:Tensor product
4527:
4522:
4517:
4515:Function space
4512:
4507:
4501:
4499:
4492:
4491:
4489:
4488:
4483:
4478:
4473:
4468:
4463:
4458:
4453:
4451:Triple product
4448:
4443:
4437:
4435:
4429:
4428:
4426:
4425:
4420:
4415:
4410:
4405:
4400:
4395:
4389:
4387:
4381:
4380:
4378:
4377:
4372:
4367:
4365:Transformation
4362:
4357:
4355:Multiplication
4352:
4347:
4342:
4337:
4331:
4329:
4323:
4322:
4315:
4313:
4311:
4310:
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4240:
4235:
4230:
4225:
4220:
4214:
4212:
4211:Basic concepts
4208:
4207:
4205:
4204:
4199:
4193:
4190:
4189:
4186:Linear algebra
4184:
4182:
4181:
4174:
4167:
4159:
4153:
4152:
4145:
4144:External links
4142:
4139:
4138:
4125:
4110:
4096:
4067:
4060:
4054:. p. 83.
4039:
4038:
4036:
4033:
4032:
4031:
4026:
4019:
4016:
3954:
3951:
3928:
3925:
3911:
3903:
3898:
3894:
3890:
3886:
3882:
3876:
3872:
3868:
3864:
3860:
3854:
3850:
3846:
3845:
3840:
3836:
3830:
3826:
3822:
3818:
3813:
3809:
3805:
3801:
3797:
3791:
3787:
3783:
3782:
3777:
3773:
3767:
3763:
3759:
3755:
3751:
3745:
3741:
3737:
3733:
3728:
3724:
3720:
3719:
3717:
3712:
3707:
3699:
3695:
3691:
3687:
3683:
3679:
3675:
3671:
3667:
3666:
3664:
3657:
3649:
3645:
3641:
3640:
3635:
3631:
3627:
3626:
3621:
3617:
3613:
3612:
3610:
3605:
3593:
3587:
3583:
3577:
3572:
3556:
3552:
3548:
3533:
3530:
3529:
3528:
3509:
3503:
3480:
3462:
3442:
3439:
3438:
3437:
3418:
3412:
3393:
3387:
3371:
3353:
3333:
3330:
3329:
3328:
3314:
3307:
3301:
3287:
3280:
3231:Main article:
3228:
3225:
3191:
3188:
3174:
3169:
3160:
3156:
3152:
3144:
3139:
3134:
3130:
3123:
3118:
3113:
3087:
3082:
3073:
3069:
3065:
3058:
3054:
3050:
3043:
3039:
3035:
3030:
3025:
2999:
2994:
2989:
2985:
2981:
2976:
2971:
2957:
2954:
2939:
2933:
2929:
2925:
2918:
2913:
2906:
2901:
2896:
2891:
2886:
2879:
2874:
2866:
2860:
2856:
2852:
2845:
2840:
2835:
2831:
2824:
2821:
2818:
2815:
2811:
2807:
2803:
2799:
2794:
2790:
2767:
2759:
2754:
2749:
2745:
2740:
2735:
2734:
2732:
2727:
2723:
2700:
2692:
2687:
2682:
2678:
2673:
2668:
2665:
2664:
2662:
2657:
2652:
2647:
2627:
2624:
2612:
2604:
2595:
2591:
2587:
2580:
2576:
2572:
2565:
2560:
2551:
2546:
2542:
2538:
2531:
2527:
2523:
2516:
2511:
2506:
2484:
2478:
2475:
2468:
2461:
2458:
2452:
2448:
2443:
2439:
2434:
2429:
2407:
2401:
2397:
2393:
2388:
2379:
2375:
2371:
2365:
2361:
2357:
2350:
2344:
2341:
2333:
2329:
2325:
2320:
2315:
2289:
2286:
2274:
2268:
2264:
2260:
2253:
2248:
2241:
2236:
2231:
2226:
2221:
2214:
2209:
2201:
2196:
2192:
2169:
2163:
2159:
2155:
2149:
2145:
2141:
2134:
2131:
2128:
2125:
2121:
2117:
2113:
2109:
2104:
2100:
2087:
2084:
2072:
2069:
2066:
2062:
2058:
2054:
2049:
2045:
2041:
2037:
2033:
2029:
2025:
1980:
1977:
1962:
1959:
1952:
1948:
1945:
1942:
1938:
1934:
1930:
1925:
1921:
1917:
1913:
1908:
1903:
1877:
1872:
1867:
1863:
1859:
1854:
1849:
1823:
1820:
1805:
1801:
1797:
1792:
1786:
1780:
1777:
1741:
1738:
1713:
1709:
1685:
1682:
1676:
1673:
1670:
1667:
1663:
1659:
1655:
1651:
1648:
1642:
1639:
1631:
1627:
1623:
1618:
1613:
1569:
1566:
1530:
1527:
1524:
1521:
1517:
1513:
1509:
1505:
1500:
1496:
1468:Main article:
1465:
1462:
1460:
1454:
1426:, the norm of
1414:
1410:
1406:
1396:is written as
1378:
1370:
1362:
1359:
1347:
1339:
1330:
1326:
1322:
1315:
1311:
1307:
1300:
1295:
1286:
1281:
1277:
1273:
1266:
1262:
1258:
1251:
1245:
1241:
1237:
1232:
1223:
1219:
1215:
1209:
1205:
1201:
1194:
1188:
1185:
1178:
1171:
1168:
1162:
1158:
1153:
1149:
1145:
1141:
1135:
1130:
1108:
1086:
1007:
1004:
998:
994:
990:
987:
984:
981:
977:
973:
969:
965:
960:
956:
899:
895:
871:
868:
860:
856:
852:
847:
842:
806:
801:
796:
791:
786:
781:
776:
772:
751:
747:
724:
719:
697:
693:
670:
665:
643:
639:
635:
629:
624:
620:
615:
610:
587:
583:
577:
572:
568:
563:
558:
533:
509:
490:
474:Projection of
457:
453:
449:
443:
438:
434:
430:
426:
422:
418:
412:
407:
367:
363:
357:
352:
330:
326:
320:
315:
274:
257:
253:
247:
242:
189:
172:
168:
162:
157:
84:
83:
77:September 2024
53:
51:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4629:
4618:
4615:
4613:
4610:
4608:
4605:
4604:
4602:
4587:
4579:
4578:
4575:
4569:
4566:
4564:
4563:Sparse matrix
4561:
4559:
4556:
4554:
4551:
4549:
4546:
4545:
4543:
4541:
4537:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4502:
4500:
4498:constructions
4497:
4493:
4487:
4486:Outermorphism
4484:
4482:
4479:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4446:Cross product
4444:
4442:
4439:
4438:
4436:
4434:
4430:
4424:
4421:
4419:
4416:
4414:
4413:Outer product
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4393:Orthogonality
4391:
4390:
4388:
4386:
4382:
4376:
4373:
4371:
4370:Cramer's rule
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4340:Decomposition
4338:
4336:
4333:
4332:
4330:
4328:
4324:
4319:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4215:
4213:
4209:
4203:
4200:
4198:
4195:
4194:
4191:
4187:
4180:
4175:
4173:
4168:
4166:
4161:
4160:
4157:
4151:
4148:
4147:
4143:
4135:
4129:
4126:
4121:
4114:
4111:
4106:
4100:
4097:
4086:
4082:
4076:
4074:
4072:
4068:
4063:
4061:9783540890676
4057:
4053:
4052:
4044:
4041:
4034:
4030:
4027:
4025:
4022:
4021:
4017:
4015:
4013:
4009:
4005:
4001:
3997:
3991:
3988:
3984:
3980:
3974:
3972:
3969:-dimensional
3968:
3964:
3960:
3952:
3950:
3948:
3944:
3941:
3937:
3934:
3926:
3924:
3909:
3901:
3896:
3892:
3884:
3880:
3874:
3870:
3862:
3858:
3852:
3848:
3838:
3834:
3828:
3824:
3816:
3811:
3807:
3799:
3795:
3789:
3785:
3775:
3771:
3765:
3761:
3753:
3749:
3743:
3739:
3731:
3726:
3722:
3715:
3710:
3705:
3697:
3693:
3685:
3681:
3673:
3669:
3662:
3655:
3647:
3643:
3633:
3629:
3619:
3615:
3608:
3603:
3581:
3570:
3559:
3544:
3539:
3531:
3524:
3517:
3508:
3504:
3499:
3492:
3486:
3479:
3475:
3474:
3473:
3470:
3461:
3455:
3449:
3440:
3433:
3426:
3417:
3413:
3408:
3401:
3392:
3388:
3383:
3377:
3370:
3366:
3365:
3364:
3361:
3352:
3346:
3340:
3331:
3324:
3313:
3306:
3302:
3297:
3286:
3279:
3275:
3274:
3273:
3269:
3264:
3260:
3256:
3252:
3247:
3241:
3234:
3226:
3221:
3216:
3210:
3205:
3201:
3196:
3189:
3187:
3172:
3154:
3142:
3132:
3121:
3116:
3101:
3098:
3085:
3067:
3052:
3041:
3033:
3028:
3012:
2997:
2987:
2979:
2974:
2955:
2953:
2950:
2937:
2916:
2904:
2894:
2889:
2877:
2864:
2843:
2833:
2822:
2819:
2816:
2813:
2797:
2792:
2788:
2765:
2757:
2743:
2730:
2725:
2698:
2690:
2676:
2666:
2660:
2655:
2650:
2634:
2625:
2623:
2610:
2589:
2574:
2563:
2549:
2525:
2514:
2509:
2482:
2466:
2450:
2441:
2437:
2432:
2405:
2359:
2348:
2331:
2327:
2323:
2318:
2302:
2296:
2287:
2285:
2272:
2251:
2239:
2229:
2224:
2212:
2199:
2194:
2190:
2180:
2167:
2143:
2132:
2129:
2126:
2123:
2107:
2102:
2098:
2085:
2083:
2070:
2067:
2064:
2035:
2027:
2014:
2010:
2006:
2001:
1995:
1978:
1976:
1950:
1946:
1943:
1940:
1923:
1919:
1911:
1906:
1890:
1875:
1865:
1857:
1852:
1836:
1830:
1821:
1819:
1784:
1763:
1758:
1711:
1707:
1671:
1668:
1665:
1646:
1629:
1625:
1621:
1616:
1600:
1594:
1588:
1582:
1576:
1567:
1565:
1563:
1558:
1555:
1549:
1544:
1528:
1525:
1522:
1519:
1503:
1498:
1494:
1484:
1478:
1471:
1463:
1459:
1455:
1453:
1451:
1446:
1440:
1435:
1432:is written ‖
1430:
1408:
1394:
1388:
1382:
1377:
1369:
1360:
1358:
1345:
1324:
1309:
1298:
1284:
1260:
1249:
1203:
1192:
1176:
1160:
1151:
1147:
1139:
1128:
1074:
1071:
1066:
1061:
1055:
1050:
1046:
1041:
1036:
1032:
1028:
1024:
996:
988:
985:
982:
979:
963:
958:
954:
944:
939:
934:
928:
922:
917:
916:
897:
893:
858:
854:
850:
845:
829:
823:
817:
804:
799:
789:
784:
774:
749:
722:
695:
668:
641:
633:
622:
618:
613:
581:
570:
566:
561:
541:
532:
525:
518:
508:
503:
497:
489:
484:
478:
472:
468:
455:
447:
436:
432:
424:
416:
405:
397:is given by:
395:
389:
383:
361:
350:
324:
313:
305:. Since both
303:
298:
294:
290:
285:
278:
273:
251:
240:
230:
226:
222:
217:
212:
207:
206:perpendicular
203:
197:
193:
188:
166:
155:
145:
139:
133:
128:
127:straight line
123:
118:
113:
107:
103:
99:
95:
91:
80:
70:
66:
60:
58:
52:
43:
42:
37:
33:
19:
4496:Vector space
4237:
4228:Vector space
4128:
4119:
4113:
4099:
4088:. Retrieved
4085:www.ck12.org
4084:
4050:
4043:
4011:
3992:
3975:
3966:
3956:
3940:vector space
3933:Gram–Schmidt
3930:
3546:
3542:
3535:
3522:
3515:
3506:
3497:
3490:
3484:
3477:
3468:
3459:
3457:is a vector
3453:
3447:
3444:
3431:
3424:
3415:
3406:
3399:
3390:
3381:
3375:
3368:
3359:
3350:
3348:is a vector
3344:
3338:
3335:
3322:
3311:
3304:
3295:
3284:
3277:
3267:
3254:
3245:
3239:
3236:
3214:
3208:
3199:
3102:
3099:
3013:
2959:
2951:
2632:
2629:
2300:
2294:
2291:
2181:
2089:
2012:
2008:
1999:
1993:
1982:
1891:
1834:
1828:
1825:
1761:
1598:
1592:
1586:
1580:
1574:
1571:
1562:scale factor
1559:
1553:
1547:
1542:
1482:
1476:
1473:
1457:
1449:
1444:
1438:
1433:
1428:
1392:
1386:
1383:
1375:
1367:
1364:
1075:
1069:
1059:
1053:
1044:
1039:
1030:
1022:
942:
932:
926:
920:
913:
827:
821:
818:
546:
539:
530:
523:
506:
501:
495:
487:
482:
476:
393:
387:
381:
301:
283:
276:
271:
228:
224:
220:
215:
210:
201:
198:
191:
186:
143:
137:
131:
129:parallel to
121:
111:
105:
97:
93:
89:
87:
74:
57:lead section
55:
4476:Multivector
4441:Determinant
4398:Dot product
4243:Linear span
2713:, which is
2005:dot product
1757:unit vector
1448:is denoted
1027:dot product
938:unit vector
708:the vector
4601:Categories
4510:Direct sum
4345:Invertible
4248:Linear map
4090:2020-09-07
4035:References
4000:hyperplane
3996:hyperplane
3190:Properties
1025:denotes a
297:orthogonal
295:) that is
293:hyperplane
4540:Numerical
4303:Transpose
3525:< 180°
3430:90° <
3321:90° <
3173:⊥
3155:⋅
3143:⊥
3133:⋅
3068:⋅
3053:⋅
3042:−
2988:−
2895:−
2844:⊥
2834:⋅
2820:θ
2817:
2667:−
2651:⊥
2590:⋅
2575:⋅
2526:⋅
2477:^
2460:^
2451:⋅
2360:⋅
2343:^
2304:becomes:
2144:⋅
2130:θ
2127:
2071:θ
2068:
2028:⋅
1961:^
1947:θ
1944:
1920:−
1866:−
1779:^
1740:^
1684:^
1672:θ
1669:
1641:^
1526:θ
1523:
1409:⋅
1325:⋅
1310:⋅
1261:⋅
1204:⋅
1187:^
1170:^
1161:⋅
1140:
1033:‖ is the
1006:^
997:⋅
986:θ
983:
870:^
634:
582:
522:90° <
448:
433:−
417:
362:
325:
287:onto the
252:
232:(denoted
167:
67:and help
4586:Category
4525:Subspace
4520:Quotient
4471:Bivector
4385:Bilinear
4327:Matrices
4202:Glossary
4018:See also
4014:-blade.
3409:< 90°
3198:If 0° ≤
2932:‖
2924:‖
2859:‖
2851:‖
2810:‖
2802:‖
2545:‖
2537:‖
2400:‖
2392:‖
2378:‖
2370:‖
2267:‖
2259:‖
2162:‖
2154:‖
2120:‖
2112:‖
2061:‖
2053:‖
2048:‖
2040:‖
2011:⋅
1937:‖
1929:‖
1804:‖
1796:‖
1662:‖
1654:‖
1516:‖
1508:‖
1361:Notation
1280:‖
1272:‖
1244:‖
1236:‖
1222:‖
1214:‖
1065:opposite
1051:between
976:‖
968:‖
4197:Outline
3521:0 <
3014:Hence,
1892:Hence,
1755:is the
1047:is the
1023:⋅
936:is the
125:onto a
115:is the
100:) of a
4481:Tensor
4293:Kernel
4223:Vector
4218:Scalar
4058:
3959:length
3500:= 180°
3434:≤ 180°
3325:≤ 180°
3263:length
3220:length
2608:
1699:where
1541:where
1343:
1043:, and
1035:length
930:, and
885:where
526:≤ 180°
102:vector
4350:Minor
4335:Block
4273:Basis
4002:. In
3983:plane
3963:angle
3943:bases
3405:0° ≤
3384:= 90°
3298:≤ 90°
3294:0° ≤
3253:<
2636:onto
2298:onto
1983:When
1049:angle
924:onto
825:onto
623:oproj
520:When
499:from
406:oproj
391:from
351:oproj
289:plane
241:oproj
141:onto
4505:Dual
4360:Rank
4056:ISBN
3961:and
3927:Uses
3493:= 0°
3422:and
3397:and
3310:= −‖
3283:= ‖
1997:and
1838:is:
1551:and
1442:and
1390:and
1129:proj
1120:as:
1098:and
1057:and
762:and
599:and
571:proj
437:proj
342:and
314:proj
225:from
156:proj
88:The
34:and
3938:of
3555:, a
3551:, a
3545:= (
3519:if
3495:or
3488:if
3451:on
3379:if
3342:on
3319:if
3292:if
3243:on
3212:on
3206:of
2814:sin
2495:or
2124:cos
2065:cos
1941:cos
1832:on
1666:cos
1590:on
1578:on
1520:cos
1480:on
1381:).
1037:of
1029:, ‖
980:cos
918:of
480:on
299:to
269:or
218:of
208:to
184:or
119:of
96:or
4603::
4083:.
4070:^
3483:=
3374:=
3257:≤
1765::
1557:.
1452:.
933:b̂
619::=
567::=
528:,
512:).
196:.
4178:e
4171:t
4164:v
4107:.
4093:.
4064:.
4012:k
3967:n
3910:]
3902:2
3897:z
3893:a
3885:z
3881:a
3875:y
3871:a
3863:z
3859:a
3853:x
3849:a
3839:z
3835:a
3829:y
3825:a
3817:2
3812:y
3808:a
3800:y
3796:a
3790:x
3786:a
3776:z
3772:a
3766:x
3762:a
3754:y
3750:a
3744:x
3740:a
3732:2
3727:x
3723:a
3716:[
3711:=
3706:]
3698:z
3694:a
3686:y
3682:a
3674:x
3670:a
3663:[
3656:]
3648:z
3644:a
3634:y
3630:a
3620:x
3616:a
3609:[
3604:=
3598:T
3592:a
3586:a
3582:=
3576:a
3571:P
3560:)
3557:z
3553:y
3549:x
3547:a
3543:a
3527:,
3523:θ
3516:b
3510:2
3507:a
3502:,
3498:θ
3491:θ
3485:0
3481:2
3478:a
3469:b
3463:2
3460:a
3454:b
3448:a
3436:.
3432:θ
3425:b
3419:1
3416:a
3411:,
3407:θ
3400:b
3394:1
3391:a
3386:,
3382:θ
3376:0
3372:1
3369:a
3360:b
3354:1
3351:a
3345:b
3339:a
3327:.
3323:θ
3317:‖
3315:1
3312:a
3308:1
3305:a
3300:,
3296:θ
3290:‖
3288:1
3285:a
3281:1
3278:a
3270:‖
3268:c
3266:‖
3255:θ
3246:b
3240:a
3215:b
3209:a
3200:θ
3168:b
3159:b
3151:b
3138:b
3129:a
3122:=
3117:2
3112:a
3086:.
3081:b
3072:b
3064:b
3057:b
3049:a
3038:a
3034:=
3029:2
3024:a
2998:1
2993:a
2984:a
2980:=
2975:2
2970:a
2938:.
2928:b
2917:y
2912:b
2905:x
2900:a
2890:x
2885:b
2878:y
2873:a
2865:=
2855:b
2839:b
2830:a
2823:=
2806:a
2798:=
2793:2
2789:a
2766:)
2758:y
2753:b
2744:x
2739:b
2731:(
2726:=
2722:b
2699:)
2691:x
2686:b
2677:y
2672:b
2661:(
2656:=
2646:b
2633:a
2611:.
2603:b
2594:b
2586:b
2579:b
2571:a
2564:=
2559:b
2550:2
2541:b
2530:b
2522:a
2515:=
2510:1
2505:a
2483:,
2474:b
2467:)
2457:b
2447:a
2442:(
2438:=
2433:1
2428:a
2406:,
2396:b
2387:b
2374:b
2364:b
2356:a
2349:=
2340:b
2332:1
2328:a
2324:=
2319:1
2314:a
2301:b
2295:a
2273:.
2263:b
2252:y
2247:b
2240:y
2235:a
2230:+
2225:x
2220:b
2213:x
2208:a
2200:=
2195:1
2191:a
2168:.
2158:b
2148:b
2140:a
2133:=
2116:a
2108:=
2103:1
2099:a
2057:b
2044:a
2036:=
2032:b
2024:a
2013:b
2009:a
2000:b
1994:a
1989:θ
1985:θ
1958:b
1951:)
1933:a
1924:(
1916:a
1912:=
1907:2
1902:a
1876:1
1871:a
1862:a
1858:=
1853:2
1848:a
1835:b
1829:a
1800:b
1791:b
1785:=
1776:b
1762:b
1737:b
1712:1
1708:a
1681:b
1675:)
1658:a
1650:(
1647:=
1638:b
1630:1
1626:a
1622:=
1617:1
1612:a
1599:b
1593:b
1587:a
1581:b
1575:a
1554:b
1548:a
1543:θ
1529:,
1512:a
1504:=
1499:1
1495:a
1483:b
1477:a
1458:θ
1450:θ
1445:b
1439:a
1434:a
1429:a
1413:b
1405:a
1393:b
1387:a
1379:1
1376:a
1371:1
1368:a
1346:.
1338:b
1329:b
1321:b
1314:b
1306:a
1299:=
1294:b
1285:2
1276:b
1265:b
1257:a
1250:=
1240:b
1231:b
1218:b
1208:b
1200:a
1193:=
1184:b
1177:)
1167:b
1157:a
1152:(
1148:=
1144:a
1134:b
1107:b
1085:a
1070:b
1060:b
1054:a
1045:θ
1040:a
1031:a
1003:b
993:a
989:=
972:a
964:=
959:1
955:a
943:b
927:b
921:a
898:1
894:a
867:b
859:1
855:a
851:=
846:1
841:a
828:b
822:a
805:.
800:2
795:a
790:+
785:1
780:a
775:=
771:a
750:,
746:b
723:2
718:a
696:,
692:b
669:1
664:a
642:.
638:a
628:b
614:2
609:a
586:a
576:b
562:1
557:a
543:.
540:b
534:1
531:a
524:θ
510:2
507:a
505:(
502:b
496:a
491:1
488:a
486:(
483:b
477:a
456:.
452:a
442:b
429:a
425:=
421:a
411:b
394:b
388:a
382:a
366:a
356:b
329:a
319:b
302:b
284:a
277:b
275:⊥
272:a
256:a
246:b
229:b
221:a
211:b
202:a
192:b
190:∥
187:a
171:a
161:b
144:b
138:a
132:b
122:a
112:b
106:a
79:)
75:(
71:.
61:.
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
