Resum Geometria afí: Plans en l'espai

2. Plans en l'espai

Plans en l'espai

Un pla de l'espai queda determinat per un punt P i dos vectors linealment independents $begin mathsize 14px style u with rightwards arrow on top comma space v with rightwards arrow on top end style$ que anomenarem vectors directors o orientadors del pla.

Tres punts A, B, C no alinats també determinen un pla ja que a partir d'aquests 3 punts també podem obtenir els dos vectors directors, per exemple, $begin mathsize 14px style stack A B with rightwards arrow on top comma space stack A C with rightwards arrow on top end style$

Equacions d'un pla

Donats

vectors orientadors $v with rightwards arrow on top equals left parenthesis u subscript 1 comma u subscript 2 comma u subscript 3 comma right parenthesis comma space space v with rightwards arrow on top equals left parenthesis v subscript 1 comma v subscript 2 comma v subscript 3 comma right parenthesis$

Punt $A equals left parenthesis a subscript 1 comma a subscript 2 comma a subscript 3 right parenthesis$

Equació vectorial

$bold left parenthesis bold italic x bold comma bold italic y bold comma bold italic z bold right parenthesis bold equals bold left parenthesis bold italic a subscript bold 1 bold comma bold italic a subscript bold 2 bold comma bold italic a subscript bold 3 bold right parenthesis bold plus bold italic lambda bold left parenthesis bold italic u subscript bold 1 bold comma bold italic u subscript bold 2 bold comma bold italic u subscript bold 3 bold right parenthesis bold plus bold italic mu bold left parenthesis bold italic v subscript bold 1 bold comma bold italic v subscript bold 2 bold comma bold italic v subscript bold 3 bold right parenthesis space space space space space space space space space$

Equació paramètrica

$open table row cell bold x bold equals bold a subscript bold 1 bold plus bold lambda bold u subscript bold 1 bold plus bold mu bold v subscript bold 1 end cell row cell bold y bold equals bold a subscript bold 2 bold plus bold lambda bold u subscript bold 2 bold plus bold mu bold v subscript bold 2 end cell row cell bold z bold equals bold a subscript bold 3 bold plus bold lambda bold u subscript bold 3 bold plus bold mu bold v subscript bold 3 end cell end table close curly brackets$

Equació general o implícita

$bold A bold x bold plus bold B bold y bold plus bold C bold z bold plus bold D bold equals bold 0$

La podem obtenir fent:

$open vertical bar table row cell x minus a subscript 1 end cell cell u subscript 1 end cell cell v subscript 1 end cell row cell y minus a subscript 2 end cell cell u subscript 2 end cell cell v subscript 2 end cell row cell z minus a subscript 3 end cell cell u subscript 3 end cell cell v subscript 3 end cell end table close vertical bar equals 0$

Exemple 1

Equacions del pla que passa pel punt $A equals left parenthesis negative 1 comma 3 comma 1 right parenthesis$ i té vectors directors $u with rightwards arrow on top equals left parenthesis 1 comma 3 comma negative 2 right parenthesis comma space space v with rightwards arrow on top equals left parenthesis 2 comma negative 1 comma 5 right parenthesis$

Equació vectorial: $left parenthesis x comma y comma z right parenthesis equals left parenthesis negative 1 comma 3 comma 1 right parenthesis plus lambda left parenthesis 1 comma 3 comma negative 2 right parenthesis plus mu left parenthesis 2 comma negative 1 comma 5 right parenthesis space space space space space space space space space$

Equació paramètrica: $open table row cell straight x equals negative 1 plus straight lambda plus 2 straight mu end cell row cell straight y equals 3 plus 3 straight lambda minus straight mu end cell row cell straight z equals 1 minus 2 straight lambda plus 5 straight mu end cell end table close curly brackets$

Equació implícita

$open vertical bar table row cell x plus 1 end cell 1 2 row cell y minus 3 end cell 3 cell negative 1 end cell row cell z minus 1 end cell cell negative 2 end cell 5 end table close vertical bar equals 0$

$15 left parenthesis x plus 1 right parenthesis minus 4 left parenthesis y minus 3 right parenthesis minus left parenthesis z minus 1 right parenthesis minus left square bracket 6 left parenthesis z minus 1 right parenthesis plus 2 left parenthesis x plus 1 right parenthesis plus 5 left parenthesis y minus 3 right parenthesis right square bracket equals 0 15 x plus 15 minus 4 y plus 12 minus z plus 1 minus 6 z plus 6 minus 2 x minus 2 minus 5 y plus 15 equals 0 13 x minus 9 y minus 7 z plus 47 equals 0$

Observació: també ho podem fer calculant el vector normal del pla: Vector normal del pla

Exemple 2

Trobeu l'equació general del pla que passa pels punts $begin mathsize 14px style A equals left parenthesis 1 comma negative 1 comma 1 right parenthesis comma space B equals left parenthesis 2 comma 0 comma 1 right parenthesis comma space C equals left parenthesis 3 comma 1 comma negative 2 right parenthesis space end style$

Trobem vectors directors:

$begin mathsize 14px style stack A B with rightwards arrow on top equals left parenthesis 1 comma 1 comma 0 right parenthesis stack A C with rightwards arrow on top equals left parenthesis 2 comma 2 comma negative 3 right parenthesis end style$

Equació general: $open vertical bar table row cell x minus 1 space end cell 1 2 row cell y plus 1 end cell 1 2 row cell z minus 1 end cell cell space 0 space end cell cell negative 3 end cell end table close vertical bar equals 0$

negative 3 left parenthesis x minus 1 right parenthesis plus 2 left parenthesis z minus 1 right parenthesis minus open square brackets 2 left parenthesis z minus 1 right parenthesis minus 3 left parenthesis y plus 1 right parenthesis close square brackets equals 0 space minus 3 x plus 3 plus 2 z minus 2 minus 2 z plus 2 plus 3 y plus 3 equals 0 space space space space space space space space space space space space space minus 3 x plus 3 y plus 6 equals 0 space

Podem simplificar:

$negative x plus y plus 2 equals 0$

Com trobar punts d'un pla?

per trobar un punt d'un pla es donen valors qualssevol a dues variables i es calcula l'altra variable. Exemple:

Volem un punt qualsevol del pla 2x-y+3z+5=0

Agafem, per exemple

$x equals 0 comma space y equals 0 space space space space rightwards double arrow space space z equals negative 5 over 3 space space space space space punt space open parentheses 0 comma 0 comma negative 5 over 3 close parentheses straight o space colon space straight x equals 0 comma space straight z equals 0 space space space rightwards double arrow space minus space straight y equals negative 5 space space space space rightwards double arrow y equals 5 space space space space space space space punt space left parenthesis 0 comma 5 comma 0 right parenthesis o colon y equals 1 comma space z equals 1 space space rightwards double arrow space 2 x minus 1 plus 3 times 1 plus 5 equals 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 2 x equals negative 7 space space rightwards double arrow x equals fraction numerator negative 7 over denominator 2 end fraction space space space space space space P u n t space open parentheses negative 7 over 2 comma 1 comma 1 close parentheses$

Propietat intel·lectual i drets d'autoria

2. Plans en l'espai