63:@0.921538:0.966074:0.940353:0.966074:0.940353:0.949450:0.921538:0.949450:0.000000:0.000000 Planos perpendiculares:@0.108233:0.097328:0.294844:0.097328:0.294844:0.079645:0.108233:0.079645:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Definición:@0.108233:0.115383:0.188317:0.115383:0.188317:0.098437:0.108233:0.098437:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.188316:0.115109:0.191212:0.115109:0.191212:0.097769:0.188316:0.097769:0.000000 Sean , , , , , :@0.191213:0.114920:0.344944:0.114920:0.344944:0.098423:0.191213:0.098423:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b c a b c , :@0.234656:0.115036:0.399814:0.115036:0.399814:0.098567:0.234656:0.098567:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000      :@0.237835:0.112896:0.346582:0.112896:0.346582:0.097078:0.237835:0.097078:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.243360:0.118266:0.248066:0.118266:0.248066:0.108665:0.243360:0.108665:0.000000 1:@0.264714:0.118266:0.269420:0.118266:0.269420:0.108665:0.264714:0.108665:0.000000 1:@0.283987:0.118266:0.288694:0.118266:0.288694:0.108665:0.283987:0.108665:0.000000 2:@0.305052:0.118266:0.309758:0.118266:0.309758:0.108665:0.305052:0.108665:0.000000 2:@0.326406:0.118266:0.331112:0.118266:0.331112:0.108665:0.326406:0.108665:0.000000 2:@0.345680:0.118266:0.350386:0.118266:0.350386:0.108665:0.345680:0.108665:0.000000 :@0.355272:0.114402:0.369925:0.114402:0.369925:0.101029:0.355272:0.101029:0.000000 R:@0.374811:0.113834:0.388115:0.113834:0.388115:0.100971:0.374811:0.100971:0.000000 3:@0.388114:0.108636:0.392820:0.108636:0.392820:0.099035:0.388114:0.099035:0.000000 tal que , , no nulos y no :@0.400566:0.114920:0.599943:0.114920:0.599943:0.098423:0.400566:0.098423:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b c:@0.455356:0.115036:0.483880:0.115036:0.483880:0.098567:0.455356:0.098567:0.000000:0.000000:0.000000  :@0.458824:0.112896:0.484516:0.112896:0.484516:0.097078:0.458824:0.097078:0.000000:0.000000:0.000000 1:@0.464339:0.118266:0.469045:0.118266:0.469045:0.108665:0.464339:0.108665:0.000000 1:@0.483612:0.118266:0.488318:0.118266:0.488318:0.108665:0.483612:0.108665:0.000000 colineales, , no nulos y no colineales, = , = , se dice :@0.108236:0.132271:0.599913:0.132271:0.599913:0.115774:0.108236:0.115774:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b c:@0.183987:0.132387:0.210723:0.132387:0.210723:0.115918:0.183987:0.115918:0.000000:0.000000:0.000000  :@0.187474:0.130247:0.211359:0.130247:0.211359:0.114429:0.187474:0.114429:0.000000:0.000000:0.000000 2:@0.192087:0.135617:0.196793:0.135617:0.196793:0.126015:0.192087:0.126015:0.000000 2:@0.209562:0.135617:0.214268:0.135617:0.214268:0.126015:0.209562:0.126015:0.000000 n:@0.395108:0.132387:0.404471:0.132387:0.404471:0.115918:0.395108:0.115918:0.000000 :@0.398674:0.130247:0.405090:0.130247:0.405090:0.114429:0.398674:0.114429:0.000000 1:@0.403298:0.135617:0.408004:0.135617:0.408004:0.126015:0.403298:0.126015:0.000000 b c n:@0.426787:0.132387:0.479349:0.132387:0.479349:0.115918:0.426787:0.115918:0.000000:0.000000:0.000000:0.000000:0.000000   :@0.430274:0.130247:0.479985:0.130247:0.479985:0.114429:0.430274:0.114429:0.000000:0.000000:0.000000:0.000000:0.000000 1 :@0.434890:0.135617:0.442011:0.135617:0.442011:0.126015:0.434890:0.126015:0.000000:0.000000 x:@0.441919:0.131630:0.448524:0.131630:0.448524:0.118001:0.441919:0.118001:0.000000 1:@0.458537:0.135617:0.463243:0.135617:0.463243:0.126015:0.458537:0.126015:0.000000 2:@0.478188:0.135617:0.482894:0.135617:0.482894:0.126015:0.478188:0.126015:0.000000 b c:@0.501677:0.132387:0.534588:0.132387:0.534588:0.115918:0.501677:0.115918:0.000000:0.000000:0.000000  :@0.505164:0.130247:0.535224:0.130247:0.535224:0.114429:0.505164:0.114429:0.000000:0.000000:0.000000 2 :@0.509780:0.135617:0.516901:0.135617:0.516901:0.126015:0.509780:0.126015:0.000000:0.000000 x:@0.516809:0.131630:0.523414:0.131630:0.523414:0.118001:0.516809:0.118001:0.000000 2 :@0.533427:0.135617:0.547950:0.135617:0.547950:0.126015:0.533427:0.126015:0.000000:0.000000:0.000000:0.000000:0.000000 que los planos: :@0.108225:0.149622:0.217227:0.149622:0.217227:0.133124:0.108225:0.133124:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.121303:0.178880:0.126236:0.178880:0.126236:0.169039:0.121303:0.169039:0.000000 =:@0.129819:0.176561:0.140672:0.176561:0.140672:0.159538:0.129819:0.159538:0.000000 x a:@0.152995:0.176681:0.187557:0.176681:0.187557:0.159687:0.152995:0.159687:0.000000:0.000000:0.000000 =:@0.164933:0.176561:0.175787:0.176561:0.175787:0.159538:0.164933:0.159538:0.000000 1:@0.186653:0.178880:0.191585:0.178880:0.191585:0.169039:0.186653:0.169039:0.000000 +:@0.193958:0.176532:0.209011:0.176532:0.209011:0.158583:0.193958:0.158583:0.000000 ub:@0.210384:0.176681:0.228557:0.176681:0.228557:0.159687:0.210384:0.159687:0.000000:0.000000 1:@0.227513:0.178880:0.232446:0.178880:0.232446:0.169039:0.227513:0.169039:0.000000 +:@0.234819:0.176532:0.249872:0.176532:0.249872:0.158583:0.234819:0.158583:0.000000 vc u v:@0.251245:0.176681:0.301011:0.176681:0.301011:0.159687:0.251245:0.159687:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.265806:0.178880:0.270739:0.178880:0.270739:0.169039:0.265806:0.169039:0.000000 | ,:@0.273112:0.176561:0.292595:0.176561:0.292595:0.159538:0.273112:0.159538:0.000000:0.000000:0.000000 {:@0.142617:0.181316:0.152291:0.181316:0.152291:0.151702:0.142617:0.151702:0.000000 }:@0.333229:0.181316:0.342903:0.181316:0.342903:0.151702:0.333229:0.151702:0.000000 ,:@0.344254:0.176561:0.347021:0.176561:0.347021:0.159538:0.344254:0.159538:0.000000 2:@0.365366:0.178880:0.370298:0.178880:0.370298:0.169039:0.365366:0.169039:0.000000 =:@0.375038:0.176561:0.385892:0.176561:0.385892:0.159538:0.375038:0.159538:0.000000 x a:@0.398215:0.176681:0.432777:0.176681:0.432777:0.159687:0.398215:0.159687:0.000000:0.000000:0.000000 =:@0.410153:0.176561:0.421007:0.176561:0.421007:0.159538:0.410153:0.159538:0.000000 2:@0.432325:0.178880:0.437258:0.178880:0.437258:0.169039:0.432325:0.169039:0.000000 +:@0.440790:0.176532:0.455842:0.176532:0.455842:0.158583:0.440790:0.158583:0.000000 xb:@0.458822:0.176681:0.475389:0.176681:0.475389:0.159687:0.458822:0.159687:0.000000:0.000000 2:@0.474799:0.178880:0.479732:0.178880:0.479732:0.169039:0.474799:0.169039:0.000000 +:@0.483264:0.176532:0.498316:0.176532:0.498316:0.158583:0.483264:0.158583:0.000000 yc x y:@0.501702:0.176681:0.555830:0.176681:0.555830:0.159687:0.501702:0.159687:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.516718:0.178880:0.521651:0.178880:0.521651:0.169039:0.516718:0.169039:0.000000 | ,:@0.525183:0.176561:0.545479:0.176561:0.545479:0.159538:0.525183:0.159538:0.000000:0.000000:0.000000 {:@0.387837:0.181316:0.397511:0.181316:0.397511:0.151702:0.387837:0.151702:0.000000 }:@0.588123:0.181316:0.597796:0.181316:0.597796:0.151702:0.588123:0.151702:0.000000 son perpendiculares si y solo si el vector normal del plano ∏ es :@0.108233:0.206854:0.599915:0.206854:0.599915:0.190357:0.108233:0.190357:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.464794:0.206970:0.474157:0.206970:0.474157:0.190501:0.464794:0.190501:0.000000 :@0.468320:0.204830:0.474735:0.204830:0.474735:0.189012:0.468320:0.189012:0.000000 1:@0.474645:0.210200:0.479351:0.210200:0.479351:0.200598:0.474645:0.200598:0.000000 1:@0.570995:0.210200:0.575701:0.210200:0.575701:0.200598:0.570995:0.200598:0.000000 ortogonal al vector normal de ∏ . Esto es, = En tal caso, se :@0.108229:0.224205:0.599914:0.224205:0.599914:0.207707:0.108229:0.207707:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.303769:0.224321:0.313132:0.224321:0.313132:0.207852:0.303769:0.207852:0.000000 :@0.307355:0.222181:0.313770:0.222181:0.313770:0.206363:0.307355:0.206363:0.000000 2:@0.312207:0.227550:0.316913:0.227550:0.316913:0.217949:0.312207:0.217949:0.000000 2:@0.355730:0.227550:0.360436:0.227550:0.360436:0.217949:0.355730:0.217949:0.000000 n n:@0.423508:0.224321:0.456716:0.224322:0.456716:0.207853:0.423508:0.207852:0.000000:0.000000:0.000000  :@0.427093:0.222181:0.457352:0.222182:0.457352:0.206364:0.427093:0.206363:0.000000:0.000000:0.000000 1 :@0.431943:0.227550:0.439064:0.227550:0.439064:0.217949:0.431943:0.217949:0.000000:0.000000 .:@0.439108:0.222153:0.443134:0.222153:0.443134:0.205207:0.439108:0.205207:0.000000 2 :@0.455790:0.227550:0.465371:0.227550:0.465371:0.217949:0.455790:0.217949:0.000000:0.000000:0.000000 :@0.480439:0.227550:0.482854:0.227550:0.482854:0.217949:0.480439:0.217949:0.000000 0. :@0.482899:0.224321:0.497964:0.224321:0.497964:0.207852:0.482899:0.207852:0.000000:0.000000:0.000000 escribe ∏:@0.108228:0.241556:0.174807:0.241556:0.174807:0.225058:0.108228:0.225058:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 :@0.174812:0.244901:0.184348:0.244901:0.184348:0.235300:0.174812:0.235300:0.000000:0.000000:0.000000 �:@0.184348:0.246068:0.194573:0.246068:0.194573:0.232250:0.184348:0.232250:0.000000 ∏ que se lee “el plano ∏ es ortogonal al plano ∏:@0.194568:0.241556:0.549614:0.241556:0.549614:0.225058:0.194568:0.225058:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.211233:0.244901:0.215940:0.244901:0.215940:0.235300:0.211233:0.235300:0.000000 , :@0.215939:0.241671:0.222933:0.241671:0.222933:0.225203:0.215939:0.225203:0.000000:0.000000 1:@0.375085:0.244901:0.379791:0.244901:0.379791:0.235300:0.375085:0.235300:0.000000 2:@0.549613:0.244901:0.554320:0.244901:0.554320:0.235300:0.549613:0.235300:0.000000 .:@0.554319:0.241671:0.557170:0.241671:0.557170:0.225203:0.554319:0.225203:0.000000 Planos paralelos:@0.108233:0.274134:0.237730:0.274134:0.237730:0.256452:0.108233:0.256452:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.237732:0.273491:0.246016:0.273491:0.246016:0.256994:0.237732:0.256994:0.000000:0.000000 Definición:@0.108231:0.291305:0.188315:0.291305:0.188315:0.274359:0.108231:0.274359:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . Sean , , , , , :@0.188317:0.290842:0.346194:0.290842:0.346194:0.274344:0.188317:0.274344:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b c a b c ,:@0.234746:0.290958:0.397267:0.290958:0.397267:0.274489:0.234746:0.274489:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000      :@0.237925:0.288818:0.347832:0.288818:0.347832:0.273000:0.237925:0.273000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.243566:0.294187:0.248272:0.294187:0.248272:0.284586:0.243566:0.284586:0.000000 1:@0.265151:0.294187:0.269857:0.294187:0.269857:0.284586:0.265151:0.284586:0.000000 1:@0.284656:0.294187:0.289362:0.294187:0.289362:0.284586:0.284656:0.284586:0.000000 2:@0.305953:0.294187:0.310659:0.294187:0.310659:0.284586:0.305953:0.284586:0.000000 2:@0.327540:0.294187:0.332246:0.294187:0.332246:0.284586:0.327540:0.284586:0.000000 2:@0.347045:0.294187:0.351751:0.294187:0.351751:0.284586:0.347045:0.284586:0.000000 :@0.356753:0.290323:0.371406:0.290323:0.371406:0.276950:0.356753:0.276950:0.000000 R:@0.376405:0.289755:0.389710:0.289755:0.389710:0.276893:0.376405:0.276893:0.000000 3:@0.389710:0.284558:0.394416:0.284558:0.394416:0.274957:0.389710:0.274957:0.000000 tal que , no nulos y no :@0.397267:0.290842:0.599940:0.290842:0.599940:0.274344:0.397267:0.274344:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b c:@0.457298:0.290958:0.486055:0.290958:0.486055:0.274489:0.457298:0.274489:0.000000:0.000000:0.000000  :@0.460765:0.288818:0.486691:0.288818:0.486691:0.273000:0.460765:0.273000:0.000000:0.000000:0.000000 1:@0.466399:0.294187:0.471105:0.294187:0.471105:0.284586:0.466399:0.284586:0.000000 1:@0.485904:0.294187:0.490610:0.294187:0.490610:0.284586:0.485904:0.284586:0.000000 colineales, , no nulos y no colineales, = , = , se dice :@0.108233:0.308193:0.599904:0.308193:0.599904:0.291695:0.108233:0.291695:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b c:@0.184003:0.308308:0.210791:0.308308:0.210791:0.291840:0.184003:0.291840:0.000000:0.000000:0.000000  :@0.187490:0.306169:0.211427:0.306169:0.211427:0.290350:0.187490:0.290350:0.000000:0.000000:0.000000 2:@0.192137:0.311538:0.196844:0.311538:0.196844:0.301937:0.192137:0.301937:0.000000 2:@0.209660:0.311538:0.214367:0.311538:0.214367:0.301937:0.209660:0.301937:0.000000 n:@0.395323:0.308308:0.404686:0.308308:0.404686:0.291840:0.395323:0.291840:0.000000 :@0.398927:0.306169:0.405343:0.306169:0.405343:0.290350:0.398927:0.290350:0.000000 1:@0.403569:0.311538:0.408276:0.311538:0.408276:0.301937:0.403569:0.301937:0.000000 b c n:@0.427097:0.308308:0.479779:0.308308:0.479779:0.291840:0.427097:0.291840:0.000000:0.000000:0.000000:0.000000:0.000000   :@0.430584:0.306169:0.480414:0.306169:0.480414:0.290350:0.430584:0.290350:0.000000:0.000000:0.000000:0.000000:0.000000 1 :@0.435236:0.311538:0.442357:0.311538:0.442357:0.301937:0.435236:0.301937:0.000000:0.000000 x:@0.442281:0.307552:0.448886:0.307552:0.448886:0.293923:0.442281:0.293923:0.000000 1:@0.458946:0.311538:0.463652:0.311538:0.463652:0.301937:0.458946:0.301937:0.000000 2:@0.478647:0.311538:0.483353:0.311538:0.483353:0.301937:0.478647:0.301937:0.000000 b c :@0.502175:0.308308:0.547965:0.308308:0.547965:0.291840:0.502175:0.291840:0.000000:0.000000:0.000000:0.000000:0.000000  :@0.505662:0.306169:0.535791:0.306169:0.535791:0.290350:0.505662:0.290350:0.000000:0.000000:0.000000 2 :@0.510314:0.311538:0.517435:0.311538:0.517435:0.301937:0.510314:0.301937:0.000000:0.000000 x:@0.517358:0.307552:0.523963:0.307552:0.523963:0.293923:0.517358:0.293923:0.000000 2 :@0.534024:0.311538:0.541145:0.311538:0.541145:0.301937:0.534024:0.301937:0.000000:0.000000 que los planos::@0.108236:0.325544:0.213094:0.325544:0.213094:0.309046:0.108236:0.309046:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.121799:0.354799:0.126715:0.354799:0.126715:0.344959:0.121799:0.344959:0.000000 =:@0.130284:0.352481:0.141101:0.352481:0.141101:0.335460:0.130284:0.335460:0.000000 x a:@0.153382:0.352601:0.187825:0.352601:0.187825:0.335609:0.153382:0.335609:0.000000:0.000000:0.000000 =:@0.165279:0.352481:0.176096:0.352481:0.176096:0.335460:0.165279:0.335460:0.000000 1:@0.186924:0.354799:0.191840:0.354799:0.191840:0.344959:0.186924:0.344959:0.000000 +:@0.194205:0.352451:0.209206:0.352451:0.209206:0.334505:0.194205:0.334505:0.000000 ub:@0.210575:0.352601:0.228685:0.352601:0.228685:0.335609:0.210575:0.335609:0.000000:0.000000 1:@0.227646:0.354799:0.232562:0.354799:0.232562:0.344959:0.227646:0.344959:0.000000 +:@0.234926:0.352451:0.249927:0.352451:0.249927:0.334505:0.234926:0.334505:0.000000 vc u v:@0.251296:0.352601:0.300890:0.352601:0.300890:0.335609:0.251296:0.335609:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.265806:0.354799:0.270722:0.354799:0.270722:0.344959:0.265806:0.344959:0.000000 | ,:@0.273087:0.352481:0.292503:0.352481:0.292503:0.335460:0.273087:0.335460:0.000000:0.000000:0.000000 {:@0.143038:0.357234:0.152678:0.357234:0.152678:0.327624:0.143038:0.327624:0.000000 }:@0.332998:0.357234:0.342638:0.357234:0.342638:0.327624:0.332998:0.327624:0.000000 ,:@0.343986:0.352481:0.346743:0.352481:0.346743:0.335460:0.343986:0.335460:0.000000 2:@0.367585:0.354799:0.372501:0.354799:0.372501:0.344959:0.367585:0.344959:0.000000 =:@0.377226:0.352481:0.388043:0.352481:0.388043:0.335460:0.377226:0.335460:0.000000 x a:@0.400323:0.352601:0.434765:0.352601:0.434765:0.335609:0.400323:0.335609:0.000000:0.000000:0.000000 =:@0.412219:0.352481:0.423036:0.352481:0.423036:0.335460:0.412219:0.335460:0.000000 2:@0.434317:0.354799:0.439233:0.354799:0.439233:0.344959:0.434317:0.344959:0.000000 +:@0.442752:0.352451:0.457753:0.352451:0.457753:0.334505:0.442752:0.334505:0.000000 xb:@0.460722:0.352601:0.477233:0.352601:0.477233:0.335609:0.460722:0.335609:0.000000:0.000000 2:@0.476645:0.354799:0.481561:0.354799:0.481561:0.344959:0.476645:0.344959:0.000000 +:@0.485080:0.352451:0.500081:0.352451:0.500081:0.334505:0.485080:0.334505:0.000000 yc x y:@0.503455:0.352601:0.557399:0.352601:0.557399:0.335609:0.503455:0.335609:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.518421:0.354799:0.523336:0.354799:0.523336:0.344959:0.518421:0.344959:0.000000 | ,:@0.526857:0.352481:0.547083:0.352481:0.547083:0.335460:0.526857:0.335460:0.000000:0.000000:0.000000 {:@0.389980:0.357234:0.399621:0.357234:0.399621:0.327624:0.389980:0.327624:0.000000 }:@0.589580:0.357234:0.599221:0.357234:0.599221:0.327624:0.589580:0.327624:0.000000 son paralelos si y solo si existe ≠:@0.108233:0.382776:0.350686:0.382776:0.350686:0.366278:0.108233:0.366278:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c 0, :@0.327853:0.382892:0.370606:0.382892:0.370606:0.366423:0.327853:0.366423:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tal que = , es decir, y :@0.371319:0.382776:0.599910:0.382776:0.599910:0.366278:0.371319:0.366278:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.430887:0.382892:0.440250:0.382892:0.440250:0.366423:0.430887:0.366423:0.000000 :@0.434471:0.380752:0.440886:0.380752:0.440886:0.364933:0.434471:0.364933:0.000000 2 :@0.439951:0.386121:0.460709:0.386121:0.460709:0.376520:0.439951:0.376520:0.000000:0.000000:0.000000:0.000000 cn:@0.461124:0.382892:0.477519:0.382892:0.477519:0.366423:0.461124:0.366423:0.000000:0.000000 :@0.471739:0.380752:0.478154:0.380752:0.478154:0.364933:0.471739:0.364933:0.000000 1:@0.477226:0.386121:0.481932:0.386121:0.481932:0.376520:0.477226:0.376520:0.000000 n:@0.550688:0.382892:0.560051:0.382892:0.560051:0.366423:0.550688:0.366423:0.000000 :@0.554273:0.380752:0.560688:0.380752:0.560688:0.364933:0.554273:0.364933:0.000000 1:@0.559755:0.386121:0.564461:0.386121:0.564461:0.376520:0.559755:0.376520:0.000000 n:@0.581993:0.382892:0.591355:0.382892:0.591355:0.366423:0.581993:0.366423:0.000000 :@0.585576:0.380752:0.591991:0.380752:0.591991:0.364933:0.585576:0.364933:0.000000 2:@0.591061:0.386121:0.595767:0.386121:0.595767:0.376520:0.591061:0.376520:0.000000 son colineales. En tal caso, se escribe ∏:@0.108241:0.400127:0.385582:0.400127:0.385582:0.383629:0.108241:0.383629:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 :@0.385527:0.403472:0.392648:0.403472:0.392648:0.393871:0.385527:0.393871:0.000000:0.000000 || ,:@0.392925:0.400242:0.431916:0.400242:0.431916:0.383774:0.392925:0.383774:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ∏ que se lee “el plano ∏ :@0.407220:0.400127:0.599910:0.400127:0.599910:0.383629:0.407220:0.383629:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.419735:0.403472:0.424442:0.403472:0.424442:0.393871:0.419735:0.393871:0.000000 1:@0.591061:0.403472:0.595767:0.403472:0.595767:0.393871:0.591061:0.393871:0.000000 es paralelo al plano ∏:@0.108241:0.417477:0.260262:0.417477:0.260262:0.400980:0.108241:0.400980:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.260253:0.420823:0.264959:0.420823:0.264959:0.411222:0.260253:0.411222:0.000000 .:@0.264958:0.417593:0.271952:0.417593:0.271952:0.401124:0.264958:0.401124:0.000000:0.000000 Ejercicios resueltos:@0.108236:0.445267:0.238891:0.445267:0.238891:0.428481:0.108236:0.428481:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.238891:0.444892:0.243033:0.444892:0.243033:0.428394:0.238891:0.428394:0.000000 1.:@0.108236:0.462459:0.120508:0.462459:0.120508:0.445745:0.108236:0.445745:0.000000:0.000000 Consideremos los planos::@0.120508:0.462242:0.311654:0.462242:0.311654:0.445745:0.120508:0.445745:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.145954:0.490642:0.151118:0.490642:0.151118:0.480616:0.145954:0.480616:0.000000 =:@0.154867:0.488279:0.166228:0.488279:0.166228:0.470936:0.154867:0.470936:0.000000 x:@0.179128:0.488400:0.186985:0.488400:0.186985:0.471088:0.179128:0.471088:0.000000 =(0, 0, 8):@0.191623:0.488279:0.264103:0.488279:0.264103:0.470936:0.191623:0.470936:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 +:@0.265764:0.488248:0.281520:0.488248:0.281520:0.469964:0.265764:0.469964:0.000000 u:@0.282958:0.488400:0.292638:0.488400:0.292638:0.471088:0.282958:0.471088:0.000000 (1, 0, 2):@0.292557:0.488279:0.369274:0.488279:0.369274:0.470936:0.292557:0.470936:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 +:@0.370935:0.488248:0.386691:0.488248:0.386691:0.469964:0.370935:0.469964:0.000000 v:@0.388129:0.488400:0.396310:0.488400:0.396310:0.471088:0.388129:0.471088:0.000000 (0, 1, 1)| , :@0.396877:0.488279:0.481732:0.488279:0.481732:0.470936:0.396877:0.470936:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u v:@0.461723:0.488400:0.489772:0.488400:0.489772:0.471088:0.461723:0.471088:0.000000:0.000000:0.000000 {:@0.168264:0.489925:0.178389:0.489925:0.178389:0.467435:0.168264:0.467435:0.000000 }:@0.523518:0.489925:0.533644:0.489925:0.533644:0.467435:0.523518:0.467435:0.000000 ,:@0.535061:0.488279:0.537957:0.488279:0.537957:0.470936:0.535061:0.470936:0.000000 2:@0.146429:0.517240:0.151592:0.517240:0.151592:0.507214:0.146429:0.507214:0.000000 =:@0.156555:0.514878:0.167916:0.514878:0.167916:0.497536:0.156555:0.497536:0.000000 x:@0.180814:0.515000:0.188672:0.515000:0.188672:0.497688:0.180814:0.497688:0.000000 =(0, 8, 0):@0.193310:0.514878:0.283836:0.514878:0.283836:0.497536:0.193310:0.497536:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 +:@0.285497:0.514848:0.301253:0.514848:0.301253:0.496563:0.285497:0.496563:0.000000 r:@0.303541:0.515000:0.309880:0.515000:0.309880:0.497688:0.303541:0.497688:0.000000 (1, 2, 0):@0.311034:0.514878:0.369703:0.514878:0.369703:0.497536:0.311034:0.497536:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 +:@0.371364:0.514848:0.387120:0.514848:0.387120:0.496563:0.371364:0.496563:0.000000 s:@0.389813:0.515000:0.396516:0.515000:0.396516:0.497688:0.389813:0.497688:0.000000 (0, 1, 1)| , :@0.396456:0.514878:0.516154:0.514878:0.516154:0.497536:0.396456:0.497536:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 r s:@0.498251:0.515000:0.522715:0.515000:0.522715:0.497688:0.498251:0.497688:0.000000:0.000000:0.000000 {:@0.169951:0.516523:0.180076:0.516523:0.180076:0.494033:0.169951:0.494033:0.000000 }:@0.555785:0.516523:0.565911:0.516523:0.565911:0.494033:0.555785:0.494033:0.000000 .:@0.566493:0.514878:0.569389:0.514878:0.569389:0.497536:0.566493:0.497536:0.000000 :@0.108233:0.538674:0.112375:0.538674:0.112375:0.522176:0.108233:0.522176:0.000000 Probemos que los planos son paralelos. Calculamos los vectores :@0.131968:0.538674:0.599877:0.538674:0.599877:0.522176:0.131968:0.522176:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 normales::@0.131968:0.556024:0.200822:0.556024:0.200822:0.539527:0.131968:0.539527:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.133462:0.587900:0.140002:0.587900:0.140002:0.574145:0.133462:0.574145:0.000000 1:@0.139769:0.587290:0.143200:0.587290:0.143200:0.579324:0.139769:0.579324:0.000000 =(1,0, 2) (0,1,1)=:@0.145693:0.587804:0.245611:0.587804:0.245611:0.574024:0.145693:0.574024:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.256565:0.581234:0.262500:0.581234:0.262500:0.567455:0.256565:0.567455:0.000000 2:@0.284854:0.581231:0.290788:0.581231:0.290788:0.567451:0.284854:0.567451:0.000000 1:@0.256849:0.597679:0.262784:0.597679:0.262784:0.583899:0.256849:0.583899:0.000000 1:@0.280359:0.597679:0.286293:0.597679:0.286293:0.583899:0.280359:0.583899:0.000000 i:@0.301010:0.587900:0.303661:0.587900:0.303661:0.574145:0.301010:0.574145:0.000000 1:@0.325329:0.581231:0.331264:0.581231:0.331264:0.567451:0.325329:0.567451:0.000000 2:@0.353331:0.581231:0.359265:0.581231:0.359265:0.567451:0.353331:0.567451:0.000000 0:@0.325044:0.597425:0.330979:0.597425:0.330979:0.583646:0.325044:0.583646:0.000000 1:@0.348836:0.597425:0.354771:0.597425:0.354771:0.583646:0.348836:0.583646:0.000000 j:@0.371448:0.587900:0.374140:0.587900:0.374140:0.574145:0.371448:0.574145:0.000000 +:@0.375303:0.587780:0.385773:0.587780:0.385773:0.573252:0.375303:0.573252:0.000000 1 0:@0.396606:0.581101:0.421003:0.581101:0.421003:0.567322:0.396606:0.567322:0.000000:0.000000:0.000000 0 1:@0.396323:0.597549:0.421286:0.597549:0.421286:0.583770:0.396323:0.583770:0.000000:0.000000:0.000000 k:@0.431234:0.587900:0.436994:0.587900:0.436994:0.574145:0.431234:0.574145:0.000000 =2:@0.441643:0.587804:0.456755:0.587804:0.456755:0.574024:0.441643:0.574024:0.000000:0.000000 i j k:@0.457199:0.587900:0.495769:0.587900:0.495769:0.574145:0.457199:0.574145:0.000000:0.000000:0.000000:0.000000:0.000000 +:@0.478019:0.587780:0.488488:0.587780:0.488488:0.573252:0.478019:0.573252:0.000000 = 2, 1,1 ,:@0.498063:0.587804:0.550556:0.587804:0.550556:0.574024:0.498063:0.574024:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (:@0.506404:0.588989:0.511639:0.588989:0.511639:0.571700:0.506404:0.571700:0.000000 ):@0.542644:0.588989:0.547879:0.588989:0.547879:0.571700:0.542644:0.571700:0.000000 n:@0.133455:0.629576:0.139996:0.629576:0.139996:0.615821:0.133455:0.615821:0.000000 2:@0.140086:0.628961:0.143517:0.628961:0.143517:0.620995:0.140086:0.620995:0.000000 =(1,2,0) (0, 1, 1)=:@0.146814:0.629473:0.257666:0.629473:0.257666:0.615694:0.146814:0.615694:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.272549:0.622903:0.278484:0.622903:0.278484:0.609124:0.272549:0.609124:0.000000 0:@0.299571:0.622903:0.305505:0.622903:0.305505:0.609124:0.299571:0.609124:0.000000 1:@0.278159:0.639348:0.284094:0.639348:0.284094:0.625569:0.278159:0.625569:0.000000 1:@0.305039:0.639348:0.310973:0.639348:0.310973:0.625569:0.305039:0.625569:0.000000 i:@0.319792:0.629570:0.322443:0.629570:0.322443:0.615815:0.319792:0.615815:0.000000 1:@0.344114:0.622774:0.350048:0.622774:0.350048:0.608995:0.344114:0.608995:0.000000 0:@0.366641:0.622774:0.372575:0.622774:0.372575:0.608995:0.366641:0.608995:0.000000 0:@0.343831:0.639222:0.349765:0.639222:0.349765:0.625443:0.343831:0.625443:0.000000 1:@0.372113:0.639222:0.378048:0.639222:0.378048:0.625443:0.372113:0.625443:0.000000 j:@0.388831:0.629570:0.391522:0.629570:0.391522:0.615815:0.388831:0.615815:0.000000 +:@0.392685:0.629449:0.403155:0.629449:0.403155:0.614921:0.392685:0.614921:0.000000 1:@0.413988:0.622903:0.419922:0.622903:0.419922:0.609124:0.413988:0.609124:0.000000 2:@0.436380:0.622903:0.442315:0.622903:0.442315:0.609124:0.436380:0.609124:0.000000 0:@0.413705:0.639098:0.419640:0.639098:0.419640:0.625319:0.413705:0.625319:0.000000 1:@0.441993:0.639097:0.447927:0.639097:0.447927:0.625317:0.441993:0.625317:0.000000 k:@0.456747:0.629570:0.462506:0.629570:0.462506:0.615815:0.456747:0.615815:0.000000 = 2:@0.465367:0.629473:0.490296:0.629473:0.490296:0.615694:0.465367:0.615694:0.000000:0.000000:0.000000 i j k:@0.490740:0.629570:0.529300:0.629570:0.529300:0.615815:0.490740:0.615815:0.000000:0.000000:0.000000:0.000000:0.000000 +:@0.494594:0.629449:0.505063:0.629449:0.505063:0.614921:0.494594:0.614921:0.000000 = 2,1, 1 .:@0.531601:0.629473:0.593347:0.629473:0.593347:0.615694:0.531601:0.615694:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 (:@0.539941:0.630659:0.545175:0.630659:0.545175:0.613371:0.539941:0.613371:0.000000 ):@0.585991:0.630659:0.591225:0.630659:0.591225:0.613371:0.585991:0.613371:0.000000 :@0.108233:0.673827:0.112375:0.673827:0.112375:0.657330:0.108233:0.657330:0.000000 Se tiene = – , luego, ∏:@0.131968:0.673827:0.317487:0.673827:0.317487:0.657330:0.131968:0.657330:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.196295:0.673943:0.205657:0.673943:0.205657:0.657474:0.196295:0.657474:0.000000 :@0.199878:0.671803:0.206293:0.671803:0.206293:0.655985:0.199878:0.655985:0.000000 2:@0.204663:0.677173:0.209369:0.677173:0.209369:0.667571:0.204663:0.667571:0.000000 n:@0.238960:0.673943:0.248323:0.673943:0.248323:0.657474:0.238960:0.657474:0.000000 :@0.242544:0.671803:0.248959:0.671803:0.248959:0.655985:0.242544:0.655985:0.000000 1:@0.247320:0.677173:0.252026:0.677173:0.252026:0.667571:0.247320:0.667571:0.000000 1 :@0.317489:0.677173:0.324610:0.677173:0.324610:0.667571:0.317489:0.667571:0.000000:0.000000 || .:@0.324609:0.673943:0.358502:0.673943:0.358502:0.657474:0.324609:0.657474:0.000000:0.000000:0.000000:0.000000:0.000000 ∏ Más aún,:@0.338422:0.673827:0.426836:0.673827:0.426836:0.657330:0.338422:0.657330:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.350943:0.677173:0.355650:0.677173:0.355650:0.667571:0.350943:0.667571:0.000000 ∏ =∏ ={ :@0.131961:0.696456:0.219761:0.696459:0.219761:0.679961:0.131961:0.679958:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 :@0.144113:0.699804:0.151010:0.699804:0.151010:0.690203:0.144113:0.690203:0.000000:0.000000 :@0.160801:0.696574:0.164943:0.696574:0.164943:0.680106:0.160801:0.680106:0.000000 2:@0.175998:0.699804:0.180704:0.699804:0.180704:0.690203:0.175998:0.690203:0.000000 x = (x, y, z) | 2x – y + z :@0.193965:0.696574:0.387289:0.696574:0.387289:0.680106:0.193965:0.680106:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.209589:0.694434:0.216005:0.694434:0.216005:0.678616:0.209589:0.678616:0.000000 :@0.277049:0.695940:0.291701:0.695940:0.291701:0.682567:0.277049:0.682567:0.000000 R:@0.294409:0.695372:0.307714:0.695372:0.307714:0.682509:0.294409:0.682509:0.000000 3:@0.307345:0.690174:0.312051:0.690174:0.312051:0.680573:0.307345:0.680573:0.000000 = 8},que se deja como ejercicio.:@0.386210:0.696459:0.595777:0.696459:0.595777:0.679961:0.386210:0.679961:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.415647:0.696574:0.419789:0.696574:0.419789:0.680106:0.415647:0.680106:0.000000 2.:@0.108231:0.731377:0.120503:0.731377:0.120503:0.714662:0.108231:0.714662:0.000000:0.000000 Consideremos los planos::@0.120504:0.731160:0.311649:0.731160:0.311649:0.714662:0.120504:0.714662:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.215525:0.762023:0.220706:0.762023:0.220706:0.751962:0.215525:0.751962:0.000000 =:@0.224470:0.759651:0.235870:0.759651:0.235870:0.742250:0.224470:0.742250:0.000000 x:@0.248812:0.759773:0.256697:0.759773:0.256697:0.742402:0.248812:0.742402:0.000000 =( , , ):@0.261350:0.759651:0.333876:0.759651:0.333876:0.742250:0.261350:0.742250:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y z:@0.282302:0.759773:0.327516:0.759773:0.327516:0.742402:0.282302:0.742402:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.367458:0.751180:0.372639:0.751180:0.372639:0.741119:0.367458:0.741119:0.000000 |2:@0.376082:0.759651:0.392136:0.759651:0.392136:0.742250:0.376082:0.742250:0.000000:0.000000 x:@0.393660:0.759773:0.401544:0.759773:0.401544:0.742402:0.393660:0.742402:0.000000 y:@0.422228:0.759773:0.430336:0.759773:0.430336:0.742402:0.422228:0.742402:0.000000 +:@0.432714:0.759621:0.448524:0.759621:0.448524:0.741274:0.432714:0.741274:0.000000 z:@0.451653:0.759773:0.458664:0.759773:0.458664:0.742402:0.451653:0.742402:0.000000 =2 ,:@0.462139:0.759651:0.499697:0.759651:0.499697:0.742250:0.462139:0.742250:0.000000:0.000000:0.000000:0.000000 {:@0.237910:0.762908:0.248071:0.762908:0.248071:0.736488:0.237910:0.736488:0.000000 }:@0.485196:0.762908:0.495357:0.762908:0.495357:0.736488:0.485196:0.736488:0.000000 2:@0.216001:0.791572:0.221182:0.791572:0.221182:0.781512:0.216001:0.781512:0.000000 =:@0.226163:0.789202:0.237563:0.789202:0.237563:0.771800:0.226163:0.771800:0.000000 x:@0.250506:0.789324:0.258391:0.789324:0.258391:0.771953:0.250506:0.771953:0.000000 =( , , ):@0.263044:0.789202:0.335570:0.789202:0.335570:0.771800:0.263044:0.771800:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x y z:@0.283995:0.789324:0.329209:0.789324:0.329209:0.771953:0.283995:0.771953:0.000000:0.000000:0.000000:0.000000:0.000000 3:@0.369150:0.780729:0.374331:0.780729:0.374331:0.770669:0.369150:0.770669:0.000000 |:@0.377776:0.789202:0.383059:0.789202:0.383059:0.771800:0.377776:0.771800:0.000000 x:@0.400003:0.789324:0.407887:0.789324:0.407887:0.771953:0.400003:0.771953:0.000000 +:@0.410489:0.789171:0.426298:0.789171:0.426298:0.770824:0.410489:0.770824:0.000000 2 =2 .:@0.428168:0.789202:0.485833:0.789202:0.485833:0.771800:0.428168:0.771800:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 z:@0.438653:0.789324:0.445664:0.789324:0.445664:0.771953:0.438653:0.771953:0.000000 {:@0.239604:0.792457:0.249764:0.792457:0.249764:0.766037:0.239604:0.766037:0.000000 }:@0.472177:0.792457:0.482338:0.792457:0.482338:0.766037:0.472177:0.766037:0.000000 :@0.108233:0.816781:0.112375:0.816781:0.112375:0.800312:0.108233:0.800312:0.000000 Probemos que estos planos son ortogonales o perpendiculares. :@0.131968:0.816665:0.599934:0.816665:0.599934:0.800168:0.131968:0.800168:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Para ello, se obtienen los vectores normales y se calcula el produc-:@0.131968:0.834016:0.595760:0.834016:0.595760:0.817518:0.131968:0.817518:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 to escalar::@0.131968:0.851367:0.202650:0.851367:0.202650:0.834869:0.131968:0.834869:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.253320:0.868833:0.262683:0.868833:0.262683:0.852365:0.253320:0.852365:0.000000 :@0.256903:0.866693:0.263318:0.866693:0.263318:0.850875:0.256903:0.850875:0.000000 = (2, –1, 1), = (–1, 0, 2),:@0.257539:0.868718:0.450672:0.868718:0.450672:0.852220:0.257539:0.852220:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.261692:0.872063:0.266398:0.872063:0.266398:0.862462:0.261692:0.862462:0.000000 n:@0.354074:0.868833:0.363437:0.868833:0.363437:0.852365:0.354074:0.852365:0.000000 :@0.357657:0.866693:0.364072:0.866693:0.364072:0.850875:0.357657:0.850875:0.000000 2:@0.362432:0.872063:0.367138:0.872063:0.367138:0.862462:0.362432:0.862462:0.000000 n n:@0.194425:0.892720:0.229174:0.892722:0.229174:0.876253:0.194425:0.876251:0.000000:0.000000:0.000000  :@0.198008:0.890580:0.229810:0.890582:0.229810:0.874764:0.198008:0.874762:0.000000:0.000000:0.000000 = (2, –1, 1) (–1, 0, 2) = –2 0 2 = 0, :@0.198644:0.892604:0.513713:0.892606:0.513713:0.876109:0.198644:0.876106:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.202795:0.895952:0.207501:0.895952:0.207501:0.886351:0.202795:0.886351:0.000000 .:@0.211643:0.890553:0.215670:0.890553:0.215670:0.873607:0.211643:0.873607:0.000000 1 :@0.228172:0.895952:0.235293:0.895952:0.235293:0.886351:0.228172:0.886351:0.000000:0.000000 . :@0.316071:0.890553:0.324124:0.890553:0.324124:0.873607:0.316071:0.873607:0.000000:0.000000 + +:@0.428040:0.892722:0.466590:0.892722:0.466590:0.876253:0.428040:0.876253:0.000000:0.000000:0.000000 :@0.108238:0.916493:0.112380:0.916493:0.112380:0.899996:0.108238:0.899996:0.000000 que muestra que es ortogonal a . Luego, ∏:@0.131973:0.916493:0.465404:0.916496:0.465404:0.899998:0.131973:0.899996:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.256752:0.916609:0.266115:0.916609:0.266115:0.900140:0.256752:0.900140:0.000000 :@0.260337:0.914469:0.266753:0.914469:0.266753:0.898651:0.260337:0.898651:0.000000 2:@0.265116:0.919841:0.269822:0.919841:0.269822:0.910239:0.265116:0.910239:0.000000 n:@0.378669:0.916611:0.388031:0.916611:0.388031:0.900143:0.378669:0.900143:0.000000 :@0.382254:0.914471:0.388669:0.914471:0.388669:0.898653:0.382254:0.898653:0.000000 1:@0.387030:0.919841:0.391736:0.919841:0.391736:0.910239:0.387030:0.910239:0.000000 1 :@0.465406:0.919841:0.474942:0.919841:0.474942:0.910239:0.465406:0.910239:0.000000:0.000000:0.000000 �:@0.474940:0.924595:0.489639:0.924595:0.489639:0.904732:0.474940:0.904732:0.000000 :@0.489633:0.919841:0.494463:0.919841:0.494463:0.910239:0.489633:0.910239:0.000000:0.000000 ∏:@0.494463:0.916496:0.506985:0.916496:0.506985:0.899998:0.494463:0.899998:0.000000 2 :@0.506985:0.919841:0.514106:0.919841:0.514106:0.910239:0.506985:0.910239:0.000000:0.000000 .:@0.514107:0.916611:0.516958:0.916611:0.516958:0.900143:0.514107:0.900143:0.000000 Interdisciplinariedad:@0.684804:0.101811:0.831589:0.101811:0.831589:0.086338:0.684804:0.086338:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Matemática y Física:@0.685042:0.130861:0.823260:0.130861:0.823260:0.115388:0.685042:0.115388:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 El producto vectorial, denomi-:@0.638498:0.146280:0.836770:0.146280:0.836770:0.131217:0.638498:0.131217:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 nado también producto cruz, :@0.638498:0.162122:0.834553:0.162122:0.834553:0.147059:0.638498:0.147059:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tiene muchas aplicaciones en :@0.638498:0.177964:0.832286:0.177964:0.832286:0.162901:0.638498:0.162901:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la física, particularmente en :@0.638498:0.193806:0.819162:0.193806:0.819162:0.178743:0.638498:0.178743:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 situaciones prácticas de nuestra :@0.638498:0.209648:0.848115:0.209648:0.848115:0.194585:0.638498:0.194585:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 vida diaria. Si posicionas los de-:@0.638498:0.225490:0.840622:0.225490:0.840622:0.210427:0.638498:0.210427:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dos de la mano derecha con el :@0.638498:0.241332:0.840288:0.241332:0.840288:0.226269:0.638498:0.226269:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 vector y rotas hacia el vector :@0.638498:0.258679:0.842645:0.258683:0.842645:0.243620:0.638498:0.243616:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.683973:0.259119:0.692950:0.259119:0.692950:0.242650:0.683973:0.242650:0.000000 :@0.687171:0.256979:0.693586:0.256979:0.693586:0.241161:0.687171:0.241161:0.000000 :@0.687807:0.259003:0.691949:0.259003:0.691949:0.242506:0.687807:0.242506:0.000000 :@0.691949:0.262349:0.694363:0.262349:0.694363:0.252748:0.691949:0.252748:0.000000 v:@0.638503:0.276470:0.646287:0.276470:0.646287:0.260001:0.638503:0.260001:0.000000 :@0.640507:0.274330:0.646922:0.274330:0.646922:0.258512:0.640507:0.258512:0.000000 :@0.641143:0.276354:0.645285:0.276354:0.645285:0.259856:0.641143:0.259856:0.000000 :@0.645285:0.279698:0.647700:0.279698:0.647700:0.270097:0.645285:0.270097:0.000000 , el dedo pulgar mostrará la :@0.647699:0.276034:0.827641:0.276034:0.827641:0.260970:0.647699:0.260970:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dirección y sentido del vector :@0.638499:0.291876:0.834378:0.291876:0.834378:0.276812:0.638499:0.276812:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 normal . A esto se conoce :@0.638499:0.309222:0.820656:0.309226:0.820656:0.294163:0.638499:0.294159:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 n:@0.689163:0.309663:0.698526:0.309663:0.698526:0.293194:0.689163:0.293194:0.000000 :@0.692746:0.307523:0.699161:0.307523:0.699161:0.291705:0.692746:0.291705:0.000000 :@0.693382:0.309547:0.697524:0.309547:0.697524:0.293049:0.693382:0.293049:0.000000 :@0.697524:0.312892:0.699939:0.312892:0.699939:0.303291:0.697524:0.303291:0.000000 como regla de la mano derecha. :@0.638496:0.325068:0.849592:0.325068:0.849592:0.310005:0.638496:0.310005:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Responde:@0.638496:0.341108:0.705285:0.341108:0.705285:0.325847:0.638496:0.325847:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 : ¿cómo saber el giro :@0.705285:0.340910:0.839988:0.340910:0.839988:0.325847:0.705285:0.325847:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que debe realizarse para abrir :@0.638496:0.356752:0.831191:0.356752:0.831191:0.341689:0.638496:0.341689:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la llave de agua o para sacar un :@0.638496:0.372594:0.842466:0.372594:0.842466:0.357531:0.638496:0.357531:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tornillo? :@0.638496:0.388436:0.694942:0.388436:0.694942:0.373373:0.638496:0.373373:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 θ:@0.775454:0.506087:0.782198:0.506087:0.782198:0.495148:0.775454:0.495148:0.000000 R:@0.713533:0.408545:0.720238:0.408545:0.720238:0.397267:0.713533:0.397267:0.000000 U:@0.825558:0.463237:0.833713:0.463237:0.833713:0.451958:0.825558:0.451958:0.000000 V:@0.825895:0.558223:0.833376:0.558223:0.833376:0.546945:0.825895:0.546945:0.000000 θ:@0.784993:0.636718:0.791737:0.636718:0.791737:0.625779:0.784993:0.625779:0.000000 – R:@0.713002:0.739592:0.729492:0.739592:0.729492:0.728313:0.713002:0.728313:0.000000:0.000000:0.000000 U:@0.831939:0.686610:0.840093:0.686610:0.840093:0.675332:0.831939:0.675332:0.000000 V:@0.832275:0.592799:0.839757:0.592799:0.839757:0.581521:0.832275:0.581521:0.000000 ©:@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