157:@0.918240:0.967835:0.943652:0.967835:0.943652:0.951847:0.918240:0.951847:0.000000:0.000000:0.000000
Longitud o norma de un vector:@0.108233:0.106129:0.358047:0.106129:0.358047:0.088446:0.108233:0.088446:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Otro de los conceptos importantes en matemática, y muy particular-:@0.108233:0.129965:0.595760:0.129965:0.595760:0.113468:0.108233:0.113468:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
mente en la geometría analítica plana, es el de la norma de un vector. :@0.108233:0.147316:0.599909:0.147316:0.599909:0.130818:0.108233:0.130818:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
En el caso del producto escalar habitual, esta norma coincide con la :@0.108233:0.164667:0.599994:0.164667:0.599994:0.148169:0.108233:0.148169:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
longitud del vector.:@0.108233:0.182018:0.245477:0.182018:0.245477:0.165520:0.108233:0.165520:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Definición:@0.108233:0.206959:0.188317:0.206959:0.188317:0.190013:0.108233:0.190013:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Sea :@0.108233:0.232855:0.136495:0.232855:0.136495:0.216358:0.108233:0.216358:0.000000:0.000000:0.000000:0.000000
(, ):@0.170198:0.232860:0.204579:0.232860:0.204579:0.216363:0.170198:0.216363:0.000000:0.000000:0.000000:0.000000
. . La longitud o norma de :@0.243225:0.232860:0.428572:0.232860:0.428572:0.216363:0.243225:0.216363:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
2:@0.235900:0.224825:0.240817:0.224825:0.240817:0.215276:0.235900:0.215276:0.000000
=:@0.156116:0.233497:0.166693:0.233497:0.166693:0.217216:0.156116:0.217216:0.000000
∈:@0.207478:0.233497:0.221214:0.233497:0.221214:0.217216:0.207478:0.217216:0.000000
A:@0.141616:0.232976:0.152886:0.232976:0.152886:0.216507:0.141616:0.216507:0.000000
a b:@0.176177:0.232976:0.199142:0.232976:0.199142:0.216507:0.176177:0.216507:0.000000:0.000000:0.000000
A:@0.433673:0.232976:0.444944:0.232976:0.444944:0.216507:0.433673:0.216507:0.000000
 que se nota :@0.448384:0.232860:0.544113:0.232860:0.544113:0.216363:0.448384:0.216363:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
A:@0.554875:0.232976:0.566146:0.232976:0.566146:0.216507:0.554875:0.216507:0.000000
 se :@0.575663:0.232860:0.599918:0.232860:0.599918:0.216363:0.575663:0.216363:0.000000:0.000000:0.000000:0.000000
define como :@0.108230:0.257122:0.202990:0.257122:0.202990:0.240625:0.108230:0.240625:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
.:@0.381135:0.258073:0.383994:0.258073:0.383994:0.240952:0.381135:0.240952:0.000000
2:@0.337619:0.251281:0.341308:0.251281:0.341308:0.244117:0.337619:0.244117:0.000000
2:@0.370705:0.248467:0.374394:0.248467:0.374394:0.241303:0.370705:0.241303:0.000000
A:@0.212721:0.258193:0.224417:0.258193:0.224417:0.241102:0.212721:0.241102:0.000000
A A:@0.262042:0.258193:0.294610:0.258193:0.294610:0.241102:0.262042:0.241102:0.000000:0.000000:0.000000
a:@0.327197:0.258193:0.336513:0.258193:0.336513:0.241102:0.327197:0.241102:0.000000
b:@0.359185:0.258193:0.368801:0.258193:0.368801:0.241102:0.359185:0.241102:0.000000
=:@0.233373:0.258733:0.244349:0.258733:0.244349:0.241838:0.233373:0.241838:0.000000
⋅ =:@0.275197:0.258733:0.310240:0.258733:0.310240:0.241838:0.275197:0.241838:0.000000:0.000000:0.000000
+:@0.346426:0.258733:0.357402:0.258733:0.357402:0.241838:0.346426:0.241838:0.000000
.:@0.370359:0.257126:0.373114:0.257126:0.373114:0.240629:0.370359:0.240629:0.000000
Nótese que para todo :@0.108236:0.283485:0.271951:0.283485:0.271951:0.266987:0.108236:0.266987:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
( ):@0.304792:0.286427:0.342417:0.286427:0.342417:0.264448:0.304792:0.264448:0.000000:0.000000:0.000000
=:@0.291127:0.284128:0.301704:0.284128:0.301704:0.267847:0.291127:0.267847:0.000000
∈:@0.344839:0.284128:0.358575:0.284128:0.358575:0.267847:0.344839:0.267847:0.000000
A:@0.276626:0.283607:0.287897:0.283607:0.287897:0.267139:0.276626:0.267139:0.000000
a b:@0.312325:0.283607:0.335308:0.283607:0.335308:0.267139:0.312325:0.267139:0.000000:0.000000:0.000000
,:@0.322227:0.283492:0.324982:0.283492:0.324982:0.266994:0.322227:0.266994:0.000000
2:@0.373268:0.275455:0.378185:0.275455:0.378185:0.265906:0.373268:0.265906:0.000000
, por la propiedad  . de la de-:@0.379330:0.283492:0.595756:0.283492:0.595756:0.266994:0.379330:0.266994:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
iv:@0.513280:0.283607:0.524800:0.283607:0.524800:0.267139:0.513280:0.267139:0.000000:0.000000
finición de producto escalar, se tiene :@0.108230:0.302722:0.372778:0.302722:0.372778:0.286225:0.108230:0.286225:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
⋅≥:@0.389590:0.303365:0.421897:0.303365:0.421897:0.287084:0.389590:0.287084:0.000000:0.000000
AA:@0.376720:0.302844:0.408103:0.302844:0.408103:0.286375:0.376720:0.286375:0.000000:0.000000
0 , con lo cual :@0.425404:0.302729:0.527604:0.302729:0.527604:0.286231:0.425404:0.286231:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
A:@0.539807:0.304102:0.551077:0.304102:0.551077:0.287633:0.539807:0.287633:0.000000
 está :@0.563002:0.302728:0.599914:0.302728:0.599914:0.286230:0.563002:0.286230:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
bien definido.:@0.108246:0.320079:0.205684:0.320079:0.205684:0.303582:0.108246:0.303581:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejercicios resueltos:@0.108242:0.344934:0.238897:0.344934:0.238897:0.328148:0.108242:0.328148:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1.:@0.108242:0.373650:0.120514:0.373650:0.120514:0.356935:0.108242:0.356935:0.000000:0.000000
   Sea  =4,3 . Entonces, :@0.120513:0.373433:0.305053:0.373437:0.305053:0.356939:0.120513:0.356935:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
( ):@0.190626:0.376372:0.225940:0.376372:0.225940:0.354393:0.190626:0.354393:0.000000:0.000000:0.000000
A:@0.163665:0.373552:0.174935:0.373552:0.174935:0.357084:0.163665:0.357084:0.000000
4:@0.364861:0.373892:0.373529:0.373892:0.373529:0.357061:0.364861:0.357061:0.000000
3:@0.409143:0.373892:0.417810:0.373892:0.417810:0.357061:0.409143:0.357061:0.000000
25,:@0.448923:0.373892:0.469993:0.373892:0.469993:0.357061:0.448923:0.357061:0.000000:0.000000:0.000000
2:@0.380991:0.363329:0.386007:0.363329:0.386007:0.353587:0.380991:0.353587:0.000000
2:@0.424845:0.363329:0.429862:0.363329:0.429862:0.353587:0.424845:0.353587:0.000000
AA:@0.308835:0.374010:0.340852:0.374010:0.340852:0.357209:0.308835:0.357209:0.000000:0.000000
() ():@0.357598:0.377636:0.424540:0.377636:0.424540:0.355711:0.357598:0.355711:0.000000:0.000000:0.000000:0.000000:0.000000
⋅=:@0.321759:0.374541:0.354661:0.374541:0.354661:0.357931:0.321759:0.357931:0.000000:0.000000
+:@0.389749:0.374541:0.400539:0.374541:0.400539:0.357931:0.389749:0.357931:0.000000
=:@0.435209:0.374541:0.446000:0.374541:0.446000:0.357931:0.435209:0.357931:0.000000
 luego  :@0.474243:0.373437:0.526009:0.373437:0.526009:0.356939:0.474243:0.356939:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
25 5.:@0.255713:0.396090:0.303105:0.396090:0.303105:0.378932:0.255713:0.378932:0.000000:0.000000:0.000000:0.000000:0.000000
A:@0.141433:0.396210:0.153154:0.396210:0.153154:0.379083:0.141433:0.379083:0.000000
A A:@0.190861:0.396210:0.223500:0.396210:0.223500:0.379083:0.190861:0.379083:0.000000:0.000000:0.000000
=:@0.162130:0.396751:0.173130:0.396751:0.173130:0.379819:0.162130:0.379819:0.000000
⋅ =:@0.204045:0.396751:0.239163:0.396751:0.239163:0.379819:0.204045:0.379819:0.000000:0.000000:0.000000
=:@0.277932:0.396751:0.288932:0.396751:0.288932:0.379819:0.277932:0.379819:0.000000
2.:@0.108233:0.422775:0.120505:0.422775:0.120505:0.406060:0.108233:0.406060:0.000000:0.000000
   Si  =0,0 , entonces :@0.120506:0.422558:0.289871:0.422558:0.289871:0.406060:0.120506:0.406060:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
( ):@0.178315:0.425493:0.214476:0.425493:0.214476:0.403514:0.178315:0.403514:0.000000:0.000000:0.000000
A:@0.151355:0.422674:0.162626:0.422674:0.162626:0.406205:0.151355:0.406205:0.000000
=0:@0.331943:0.424667:0.356664:0.424667:0.356664:0.406433:0.331943:0.406433:0.000000:0.000000
AA:@0.293977:0.424795:0.328664:0.424795:0.328664:0.406593:0.293977:0.406593:0.000000:0.000000
⋅:@0.307988:0.425371:0.313311:0.425371:0.313311:0.407376:0.307988:0.407376:0.000000
, y  :@0.360246:0.422558:0.383249:0.422558:0.383249:0.406060:0.360246:0.406060:0.000000:0.000000:0.000000:0.000000:0.000000
00.:@0.507908:0.423749:0.547924:0.423749:0.547924:0.406564:0.507908:0.406564:0.000000:0.000000:0.000000
A:@0.393009:0.423869:0.404748:0.423869:0.404748:0.406714:0.393009:0.406714:0.000000
A A:@0.442517:0.423869:0.475207:0.423869:0.475207:0.406714:0.442517:0.406714:0.000000:0.000000:0.000000
=:@0.413739:0.424412:0.424756:0.424412:0.424756:0.407453:0.413739:0.407453:0.000000
⋅ =:@0.455721:0.424412:0.490897:0.424412:0.490897:0.407453:0.455721:0.407453:0.000000:0.000000:0.000000
=:@0.521822:0.424412:0.532839:0.424412:0.532839:0.407453:0.521822:0.407453:0.000000
Propiedades de la norma de un vector:@0.108233:0.459411:0.412745:0.459411:0.412745:0.441729:0.108233:0.441729:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Se verifican las siguientes propiedades::@0.108233:0.483248:0.379966:0.483248:0.379966:0.466750:0.108233:0.466750:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
i):@0.108233:0.509824:0.118771:0.509824:0.118771:0.493109:0.108233:0.493109:0.000000:0.000000
  :@0.118771:0.509606:0.135821:0.509606:0.135821:0.493108:0.118771:0.493108:0.000000:0.000000
0,:@0.172812:0.508181:0.184565:0.508181:0.184565:0.492862:0.172812:0.492862:0.000000:0.000000
.:@0.256911:0.508181:0.259469:0.508181:0.259469:0.492862:0.256911:0.492862:0.000000
2:@0.249960:0.500720:0.254526:0.500720:0.254526:0.491853:0.249960:0.491853:0.000000
A:@0.140388:0.508288:0.150853:0.508288:0.150853:0.492996:0.140388:0.492996:0.000000
A:@0.206453:0.508288:0.216918:0.508288:0.216918:0.492996:0.206453:0.492996:0.000000
≥:@0.159929:0.508772:0.169751:0.508772:0.169751:0.493654:0.159929:0.493654:0.000000
∀ ∈:@0.193811:0.508772:0.232810:0.508772:0.232810:0.493654:0.193811:0.493654:0.000000:0.000000:0.000000
.:@0.255441:0.509612:0.258196:0.509612:0.258196:0.493115:0.255441:0.493115:0.000000
ii):@0.108236:0.537446:0.123031:0.537446:0.123031:0.520731:0.108236:0.520731:0.000000:0.000000:0.000000
  :@0.123030:0.537229:0.135823:0.537229:0.135823:0.520731:0.123030:0.520731:0.000000:0.000000
=:@0.161213:0.536613:0.103841:0.536613:0.103841:0.520332:0.161213:0.520332:0.000000
= =:@0.229161:0.536613:0.104891:0.536613:0.104891:0.520332:0.229161:0.520332:0.000000:0.000000:0.000000
AA:@0.141061:0.536092:0.153382:0.536092:0.153382:0.519624:0.141061:0.519624:0.000000:0.000000
AA:@0.214674:0.536092:0.226994:0.536092:0.226994:0.519624:0.214674:0.519624:0.000000:0.000000
0  0:@0.175719:0.535977:0.185266:0.535977:0.185266:0.519479:0.175719:0.519479:0.000000:0.000000:0.000000:0.000000
0..:@0.243668:0.535977:0.255900:0.535977:0.255900:0.519479:0.243668:0.519479:0.000000:0.000000:0.000000
⇔:@0.189778:0.537783:0.209852:0.537783:0.209852:0.518943:0.189778:0.518943:0.000000
 :@0.209852:0.537234:0.213994:0.537234:0.213994:0.520736:0.209852:0.520736:0.000000
=:@0.230211:0.536613:0.172840:0.536613:0.172840:0.520332:0.230211:0.520332:0.000000
0:@0.244718:0.535977:0.253214:0.535977:0.253214:0.519479:0.244718:0.519479:0.000000
iii):@0.108233:0.563817:0.127286:0.563817:0.127286:0.547102:0.108233:0.547102:0.000000:0.000000:0.000000:0.000000
  :@0.127287:0.563600:0.135822:0.563600:0.135822:0.547102:0.127287:0.547102:0.000000:0.000000
,:@0.228862:0.562814:0.231509:0.562814:0.231509:0.546963:0.228862:0.546963:0.000000
,:@0.299477:0.562814:0.302123:0.562814:0.302123:0.546963:0.299477:0.546963:0.000000
.:@0.371461:0.562814:0.374108:0.562814:0.374108:0.546963:0.371461:0.546963:0.000000
2:@0.364265:0.555093:0.368990:0.555093:0.368990:0.545919:0.364265:0.545919:0.000000
A:@0.152309:0.562925:0.163137:0.562925:0.163137:0.547102:0.152309:0.547102:0.000000
A:@0.210300:0.562925:0.221128:0.562925:0.221128:0.547102:0.210300:0.547102:0.000000
A:@0.319248:0.562925:0.330076:0.562925:0.330076:0.547102:0.319248:0.547102:0.000000
α:@0.137612:0.563425:0.149292:0.563425:0.153877:0.547783:0.142197:0.547783:0.000000
α:@0.185293:0.563425:0.196973:0.563425:0.201558:0.547783:0.189878:0.547783:0.000000
α:@0.250188:0.563425:0.261868:0.563425:0.266453:0.547783:0.254773:0.547783:0.000000
=:@0.171421:0.563425:0.181583:0.563425:0.181583:0.547783:0.171421:0.547783:0.000000
∀ ∈:@0.238316:0.563425:0.280999:0.563425:0.280999:0.547783:0.238316:0.547783:0.000000:0.000000:0.000000
∀ ∈:@0.306172:0.563425:0.346542:0.563425:0.346542:0.547783:0.306172:0.547783:0.000000:0.000000:0.000000
.:@0.369885:0.563600:0.372640:0.563600:0.372640:0.547102:0.369885:0.547102:0.000000
iv):@0.108225:0.590175:0.127028:0.590175:0.127028:0.573461:0.108225:0.573461:0.000000:0.000000:0.000000
  :@0.127028:0.589958:0.135813:0.589958:0.135813:0.573460:0.127028:0.573460:0.000000:0.000000
,:@0.243887:0.590445:0.246708:0.590445:0.246708:0.573551:0.243887:0.573551:0.000000
,:@0.280502:0.590445:0.283323:0.590445:0.283323:0.573551:0.280502:0.573551:0.000000
.:@0.339309:0.590445:0.342130:0.590445:0.342130:0.573551:0.339309:0.573551:0.000000
2:@0.331643:0.582216:0.336679:0.582216:0.336679:0.572438:0.331643:0.572438:0.000000
AB:@0.138602:0.590563:0.167424:0.590563:0.167424:0.573699:0.138602:0.573699:0.000000:0.000000
AB:@0.197785:0.590563:0.234537:0.590563:0.234537:0.573699:0.197785:0.573699:0.000000:0.000000
AB:@0.267897:0.590563:0.294095:0.590563:0.294095:0.573699:0.267897:0.573699:0.000000:0.000000
⋅≤:@0.151563:0.591096:0.186678:0.591096:0.186678:0.574425:0.151563:0.574425:0.000000:0.000000
∀:@0.253966:0.591096:0.268031:0.591096:0.268031:0.574425:0.253966:0.574425:0.000000
∈:@0.298668:0.591096:0.312734:0.591096:0.312734:0.574425:0.298668:0.574425:0.000000
 :@0.338726:0.589964:0.342868:0.589964:0.342868:0.573466:0.338726:0.573466:0.000000
(desigualdad de Cauchy-Schwarz).:@0.342867:0.589323:0.543535:0.589323:0.543535:0.575694:0.342867:0.575694:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
v):@0.108233:0.616545:0.122855:0.616545:0.122855:0.599831:0.108233:0.599831:0.000000:0.000000
  :@0.122856:0.616328:0.135822:0.616328:0.135822:0.599831:0.122856:0.599831:0.000000:0.000000
,:@0.266701:0.616328:0.269456:0.616328:0.269456:0.599831:0.266701:0.599831:0.000000
,:@0.302457:0.616328:0.305212:0.616328:0.305212:0.599831:0.302457:0.599831:0.000000
.  :@0.359886:0.616328:0.362974:0.616328:0.362974:0.599831:0.359886:0.599831:0.000000:0.000000:0.000000
2:@0.352399:0.608293:0.357317:0.608293:0.357317:0.598744:0.352399:0.598744:0.000000
AB:@0.141058:0.616444:0.175215:0.616444:0.175215:0.599975:0.141058:0.599975:0.000000:0.000000
A:@0.207484:0.616444:0.218754:0.616444:0.218754:0.599975:0.207484:0.599975:0.000000
B:@0.248365:0.616444:0.257555:0.616444:0.257555:0.599975:0.248365:0.599975:0.000000
AB:@0.290152:0.616444:0.315736:0.616444:0.315736:0.599975:0.290152:0.599975:0.000000:0.000000
+:@0.153715:0.616965:0.164292:0.616965:0.164292:0.600684:0.153715:0.600684:0.000000
≤:@0.186061:0.616965:0.196638:0.616965:0.196638:0.600684:0.186061:0.600684:0.000000
+:@0.228098:0.616965:0.238675:0.616965:0.238675:0.600684:0.228098:0.600684:0.000000
∀:@0.276531:0.616965:0.290267:0.616965:0.290267:0.600684:0.276531:0.600684:0.000000
∈:@0.320186:0.616965:0.333922:0.616965:0.333922:0.600684:0.320186:0.600684:0.000000
(desigualdad triangular).:@0.362975:0.615687:0.504537:0.615687:0.504537:0.602059:0.362975:0.602059:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
vi):@0.108233:0.642911:0.127094:0.642911:0.127094:0.626196:0.108233:0.626196:0.000000:0.000000:0.000000
  :@0.127095:0.642694:0.135822:0.642694:0.135822:0.626196:0.127095:0.626196:0.000000:0.000000
,:@0.267538:0.642694:0.270293:0.642694:0.270293:0.626196:0.267538:0.626196:0.000000
,:@0.306165:0.642694:0.308920:0.642694:0.308920:0.626196:0.306165:0.626196:0.000000
.:@0.363595:0.642694:0.366350:0.642694:0.366350:0.626196:0.363595:0.626196:0.000000
2:@0.356117:0.634657:0.361034:0.634657:0.361034:0.625108:0.356117:0.625108:0.000000
A:@0.141058:0.642810:0.152328:0.642810:0.152328:0.626341:0.141058:0.626341:0.000000
B:@0.182362:0.642810:0.191552:0.642810:0.191552:0.626341:0.182362:0.626341:0.000000
AB:@0.223821:0.642810:0.258402:0.642810:0.258402:0.626341:0.223821:0.626341:0.000000:0.000000
AB:@0.293870:0.642810:0.319454:0.642810:0.319454:0.626341:0.293870:0.626341:0.000000:0.000000
−:@0.162095:0.643330:0.172672:0.643330:0.172672:0.627049:0.162095:0.627049:0.000000
≤:@0.202398:0.643330:0.212975:0.643330:0.212975:0.627049:0.202398:0.627049:0.000000
−:@0.236902:0.643330:0.247479:0.643330:0.247479:0.627049:0.236902:0.627049:0.000000
∀:@0.280249:0.643330:0.293985:0.643330:0.293985:0.627049:0.280249:0.627049:0.000000
∈:@0.323904:0.643330:0.337640:0.643330:0.337640:0.627049:0.323904:0.627049:0.000000
Demostración.:@0.108230:0.667636:0.221625:0.667636:0.221625:0.650690:0.108230:0.650690:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
  Sea :@0.108233:0.693530:0.136495:0.693530:0.136495:0.677033:0.108233:0.677033:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
∈:@0.159680:0.694173:0.173416:0.694173:0.173416:0.677892:0.159680:0.677892:0.000000
A:@0.145597:0.693652:0.156867:0.693652:0.156867:0.677184:0.145597:0.677184:0.000000
2:@0.188107:0.685501:0.193024:0.685501:0.193024:0.675952:0.188107:0.675952:0.000000
. Por  la propiedad  . de la definición del  producto :@0.193396:0.693537:0.599937:0.693537:0.599937:0.677039:0.193396:0.677039:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
iv:@0.343769:0.693652:0.355289:0.693652:0.355289:0.677184:0.343769:0.677184:0.000000:0.000000
 :@0.595770:0.693537:0.599912:0.693537:0.599912:0.677039:0.595770:0.677039:0.000000
escalar, se tiene que :@0.108243:0.715283:0.251399:0.715283:0.251399:0.698785:0.108243:0.698785:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0:@0.304690:0.713461:0.313410:0.713461:0.313410:0.696530:0.304690:0.696530:0.000000
AA:@0.255200:0.713580:0.287409:0.713580:0.287409:0.696678:0.255200:0.696678:0.000000:0.000000
⋅≥:@0.268207:0.714114:0.301305:0.714114:0.301305:0.697405:0.268207:0.697405:0.000000:0.000000
, luego :@0.317940:0.715288:0.368316:0.715288:0.368316:0.698790:0.317940:0.698790:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
0.:@0.481676:0.716202:0.493303:0.716202:0.493303:0.699035:0.481676:0.699035:0.000000:0.000000
A:@0.378077:0.716322:0.389804:0.716322:0.389804:0.699186:0.378077:0.699186:0.000000
A A:@0.429936:0.716322:0.462572:0.716322:0.462572:0.699186:0.429936:0.699186:0.000000:0.000000:0.000000
=:@0.399967:0.716864:0.410973:0.716864:0.410973:0.699923:0.399967:0.699923:0.000000
⋅ ≥:@0.443107:0.716864:0.478248:0.716864:0.478248:0.699923:0.443107:0.699923:0.000000:0.000000:0.000000
 :@0.499815:0.715288:0.503957:0.715288:0.503957:0.698790:0.499815:0.698790:0.000000
El resto de propiedades se deja al estudiante como investigación.:@0.108230:0.739767:0.553244:0.739767:0.553244:0.723269:0.108230:0.723269:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Ejercicio resuelto:@0.108230:0.764622:0.226208:0.764622:0.226208:0.747835:0.108230:0.747835:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
1.:@0.108230:0.792080:0.120502:0.792080:0.120502:0.775365:0.108230:0.775365:0.000000:0.000000
   Sean :@0.120501:0.791863:0.169473:0.791863:0.169473:0.775365:0.120501:0.775365:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
=:@0.187703:0.792504:0.198280:0.792504:0.198280:0.776223:0.187703:0.776223:0.000000
= −−:@0.254920:0.792504:0.310115:0.792504:0.310115:0.776223:0.254920:0.776223:0.000000:0.000000:0.000000:0.000000
A:@0.173196:0.791983:0.184467:0.791983:0.184467:0.775515:0.173196:0.775515:0.000000
B:@0.241415:0.791983:0.250604:0.791983:0.250604:0.775515:0.241415:0.775515:0.000000
(3,4),:@0.201786:0.791868:0.237735:0.791868:0.237735:0.775370:0.201786:0.775370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
(2,:@0.269002:0.791868:0.298040:0.791868:0.298040:0.775370:0.269002:0.775370:0.000000:0.000000:0.000000
5). Verifiquemos las desigualdades de :@0.323297:0.791868:0.594083:0.791868:0.594083:0.775370:0.323297:0.775370:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
Cauchy-Schwarz. Tenemos: :@0.131701:0.809218:0.328370:0.809219:0.328370:0.792721:0.131701:0.792721:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 :@0.108250:0.835996:0.112392:0.835996:0.112392:0.819499:0.108250:0.819499:0.000000
3( 2) 4(:@0.183187:0.830632:0.282389:0.830632:0.282389:0.813422:0.183187:0.813422:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
5):@0.305943:0.830632:0.321981:0.830632:0.321981:0.813422:0.305943:0.813422:0.000000:0.000000
64 5,:@0.350078:0.830632:0.411129:0.830632:0.411129:0.813422:0.350078:0.813422:0.000000:0.000000:0.000000:0.000000:0.000000
AB:@0.135565:0.830753:0.164927:0.830753:0.164927:0.813573:0.135565:0.813573:0.000000:0.000000
⋅= ×− +× −:@0.148789:0.831296:0.293030:0.831296:0.293030:0.814312:0.148789:0.814312:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
=−−:@0.325206:0.831296:0.372275:0.831296:0.372275:0.814312:0.325206:0.814312:0.000000:0.000000:0.000000
 :@0.108233:0.869066:0.112375:0.869066:0.112375:0.852568:0.108233:0.852568:0.000000
64 5:@0.202839:0.858490:0.257834:0.858490:0.257834:0.841451:0.202839:0.841451:0.000000:0.000000:0.000000:0.000000
6 45,:@0.279879:0.858490:0.339881:0.858490:0.339881:0.841451:0.279879:0.841451:0.000000:0.000000:0.000000:0.000000:0.000000
AB:@0.138664:0.858609:0.167734:0.858609:0.167734:0.841600:0.138664:0.841600:0.000000:0.000000
⋅= −−:@0.151756:0.859147:0.224798:0.859147:0.224798:0.842332:0.151756:0.842332:0.000000:0.000000:0.000000:0.000000:0.000000
=+:@0.265566:0.859147:0.301420:0.859147:0.301420:0.842332:0.265566:0.842332:0.000000:0.000000
 :@0.108233:0.900876:0.112375:0.900876:0.112375:0.884378:0.108233:0.884378:0.000000
=:@0.161213:0.892711:0.171789:0.892711:0.171789:0.876430:0.161213:0.876430:0.000000
+:@0.206505:0.892711:0.217082:0.892711:0.217082:0.876430:0.206505:0.876430:0.000000
=:@0.241106:0.892711:0.251682:0.892711:0.251682:0.876430:0.241106:0.876430:0.000000
=:@0.289847:0.892711:0.300423:0.892711:0.300423:0.876430:0.289847:0.876430:0.000000
=:@0.347893:0.892711:0.358469:0.892711:0.358469:0.876430:0.347893:0.876430:0.000000
−:@0.380971:0.892711:0.391548:0.892711:0.391548:0.876430:0.380971:0.876430:0.000000
+−:@0.415475:0.892711:0.445317:0.892711:0.445317:0.876430:0.415475:0.876430:0.000000:0.000000
=:@0.486000:0.892711:0.496576:0.892711:0.496576:0.876430:0.486000:0.876430:0.000000
=:@0.527150:0.892711:0.537727:0.892711:0.537727:0.876430:0.527150:0.876430:0.000000
A:@0.141058:0.892190:0.152328:0.892190:0.152328:0.875721:0.141058:0.875721:0.000000
B:@0.328740:0.892190:0.337929:0.892190:0.337929:0.875721:0.328740:0.875721:0.000000
3:@0.188335:0.892075:0.196831:0.892075:0.196831:0.875577:0.188335:0.875577:0.000000
4:@0.220219:0.892075:0.228715:0.892075:0.228715:0.875577:0.220219:0.875577:0.000000
25 5,:@0.268228:0.892075:0.315755:0.892075:0.315755:0.875577:0.268228:0.875577:0.000000:0.000000:0.000000:0.000000:0.000000
(2):@0.375034:0.892075:0.405705:0.892075:0.405705:0.875577:0.375034:0.875577:0.000000:0.000000:0.000000
(:@0.428804:0.892075:0.434757:0.892075:0.434757:0.875577:0.428804:0.875577:0.000000
5):@0.458549:0.892075:0.473923:0.892075:0.473923:0.875577:0.458549:0.875577:0.000000:0.000000
93.:@0.513537:0.892075:0.552414:0.892075:0.552414:0.875577:0.513537:0.875577:0.000000:0.000000:0.000000
2:@0.197021:0.884039:0.201938:0.884039:0.201938:0.874490:0.197021:0.874490:0.000000
2:@0.229324:0.884039:0.234242:0.884039:0.234242:0.874490:0.229324:0.874490:0.000000
2:@0.406008:0.884039:0.410925:0.884039:0.410925:0.874490:0.406008:0.874490:0.000000
2:@0.474217:0.884039:0.479134:0.884039:0.479134:0.874490:0.474217:0.874490:0.000000
Entonces,:@0.131680:0.922627:0.199684:0.922627:0.199684:0.906129:0.131680:0.906129:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
64 5:@0.261466:0.922046:0.315988:0.922046:0.315988:0.905024:0.261466:0.905024:0.000000:0.000000:0.000000:0.000000
15.:@0.398616:0.922046:0.418831:0.922046:0.418831:0.905024:0.398616:0.905024:0.000000:0.000000:0.000000
AB:@0.210819:0.922165:0.239859:0.922165:0.239859:0.905173:0.210819:0.905173:0.000000:0.000000
AB:@0.341470:0.922165:0.375519:0.922165:0.375519:0.905173:0.341470:0.905173:0.000000:0.000000
⋅= +:@0.223878:0.922702:0.282972:0.922702:0.282972:0.905904:0.223878:0.905904:0.000000:0.000000:0.000000:0.000000
≤:@0.320559:0.922702:0.331472:0.922702:0.331472:0.905904:0.320559:0.905904:0.000000
=:@0.385537:0.922702:0.396449:0.922702:0.396449:0.905904:0.385537:0.905904:0.000000
En las figuras siguientes, :@0.685042:0.140456:0.841679:0.140456:0.841679:0.125393:0.685042:0.125393:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
se muestran el vector :@0.638498:0.159189:0.781259:0.159189:0.781259:0.144126:0.638498:0.144126:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
A:@0.784978:0.159878:0.796248:0.159878:0.796248:0.143409:0.784978:0.143409:0.000000
 y el :@0.799690:0.159190:0.828854:0.159190:0.828854:0.144127:0.799690:0.144127:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
triángulo rectángulo que se :@0.638513:0.175032:0.820022:0.175032:0.820022:0.159969:0.638513:0.159969:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
forma con el eje  . Obsérvese :@0.638513:0.190874:0.833281:0.190874:0.833281:0.175811:0.638513:0.175811:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
X:@0.745917:0.190980:0.755011:0.190980:0.755011:0.175943:0.745917:0.175943:0.000000
que en el punto de abscisa  , se :@0.638513:0.206716:0.846161:0.206716:0.846161:0.191653:0.638513:0.191653:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
a:@0.814587:0.206822:0.822784:0.206822:0.822784:0.191785:0.814587:0.191785:0.000000
tiene el ángulo recto. :@0.638513:0.222558:0.777437:0.222558:0.777437:0.207495:0.638513:0.207495:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
x:@0.821151:0.435356:0.828464:0.435356:0.828464:0.419545:0.821151:0.419545:0.000000
y:@0.755500:0.247716:0.763048:0.247716:0.763048:0.231905:0.755500:0.231905:0.000000
0:@0.776875:0.434594:0.784914:0.434594:0.784914:0.418782:0.776875:0.418782:0.000000
A:@0.670051:0.274954:0.680938:0.274954:0.680938:0.259142:0.670051:0.259142:0.000000
a:@0.674253:0.434601:0.683018:0.434601:0.683018:0.418789:0.674253:0.418789:0.000000
b:@0.775979:0.282519:0.784943:0.282519:0.784943:0.266708:0.775979:0.266708:0.000000
y:@0.654762:0.469572:0.661496:0.469572:0.661496:0.455467:0.654762:0.455467:0.000000
x:@0.823750:0.563768:0.830274:0.563768:0.830274:0.549662:0.823750:0.549662:0.000000
0:@0.671743:0.566428:0.678914:0.566428:0.678914:0.552323:0.671743:0.552323:0.000000
A:@0.712294:0.528473:0.722007:0.528473:0.722007:0.514368:0.712294:0.514368:0.000000
b:@0.655928:0.506133:0.663924:0.506133:0.663924:0.492028:0.655928:0.492028:0.000000
a:@0.792038:0.566431:0.799857:0.566431:0.799857:0.552326:0.792038:0.552326:0.000000
=:@0.674543:0.613748:0.685119:0.613748:0.685119:0.597468:0.674543:0.597468:0.000000
∀ ∈:@0.715539:0.613748:0.754551:0.613748:0.754551:0.597468:0.715539:0.597468:0.000000:0.000000:0.000000
a:@0.653659:0.613228:0.662637:0.613228:0.662637:0.596759:0.653659:0.596759:0.000000
a:@0.691766:0.613228:0.700744:0.613228:0.700744:0.596759:0.691766:0.596759:0.000000
a:@0.729141:0.613228:0.738118:0.613228:0.738118:0.596759:0.729141:0.596759:0.000000
||:@0.684484:0.613112:0.707236:0.613112:0.707236:0.596614:0.684484:0.596614:0.000000:0.000000
.:@0.769289:0.613112:0.772044:0.613112:0.772044:0.596614:0.769289:0.596614:0.000000
2:@0.662764:0.605076:0.667682:0.605076:0.667682:0.595528:0.662764:0.595528:0.000000
 :@0.775005:0.612792:0.778787:0.612792:0.778787:0.597728:0.775005:0.597728:0.000000
Si :@0.637627:0.629267:0.652192:0.629267:0.652192:0.614204:0.637627:0.614204:0.000000:0.000000:0.000000
=:@0.667800:0.630219:0.678377:0.630219:0.678377:0.613938:0.667800:0.613938:0.000000
u:@0.654346:0.629698:0.663555:0.629698:0.663555:0.613230:0.654346:0.613230:0.000000
a:@0.687848:0.629698:0.696826:0.629698:0.696826:0.613230:0.687848:0.613230:0.000000
(,0):@0.681876:0.629583:0.716245:0.629583:0.716245:0.613085:0.681876:0.613085:0.000000:0.000000:0.000000:0.000000
 con :@0.719210:0.629262:0.751382:0.629262:0.751382:0.614199:0.719210:0.614199:0.000000:0.000000:0.000000:0.000000:0.000000
∈:@0.766039:0.630219:0.779776:0.630219:0.779776:0.613938:0.766039:0.613938:0.000000
a:@0.754365:0.629698:0.763342:0.629698:0.763342:0.613230:0.754365:0.613230:0.000000
, :@0.795926:0.629262:0.802223:0.629262:0.802223:0.614199:0.795926:0.614199:0.000000:0.000000
entonces :@0.637633:0.650768:0.700498:0.650768:0.700498:0.635705:0.637633:0.635705:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
=:@0.727410:0.651719:0.689554:0.651719:0.689554:0.635438:0.727410:0.635438:0.000000
=:@0.775843:0.651719:0.737987:0.651719:0.737987:0.635438:0.775843:0.635438:0.000000
u:@0.708302:0.651198:0.717511:0.651198:0.717511:0.634730:0.708302:0.634730:0.000000
a:@0.754943:0.651198:0.763921:0.651198:0.763921:0.634730:0.754943:0.634730:0.000000
a:@0.793262:0.651198:0.802240:0.651198:0.802240:0.634730:0.793262:0.634730:0.000000
.:@0.806459:0.651083:0.809214:0.651083:0.809214:0.634585:0.806459:0.634585:0.000000
2:@0.764056:0.643047:0.768973:0.643047:0.768973:0.633498:0.764056:0.633498:0.000000
Competencia :@0.684804:0.103922:0.782426:0.103922:0.782426:0.088449:0.684804:0.088449:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
matemática:@0.684804:0.116490:0.768866:0.116490:0.768866:0.101017:0.684804:0.101017:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000
 ©:@0.079553:0.731952:0.079553:0.720342:0.061505:0.720342:0.061505:0.731952:0.000000:0.000000
maya:@0.080774:0.720340:0.080774:0.684191:0.055786:0.684191:0.055786:0.720340:0.000000:0.000000:0.000000:0.000000
®EDUCACIÓN – Libro resuelto solo para fines didácticos – Prohibida su reproducción :@0.079553:0.684191:0.079553:0.268044:0.061505:0.268044:0.061505:0.684191:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000