Dubtes freqüents sobre geometria: Equacions d'un pla

GEOMETRIA AFÍ

Equacions d'un pla

Donats

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

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

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

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

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

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

$15 left parenthesis x plus 1 right parenthesis minus 4 left parenthesis y minus 3 right parenthesis minus left parenthesis z minus 1 right parenthesis minus left square bracket 6 left parenthesis z minus 1 right parenthesis plus 2 left parenthesis x plus 1 right parenthesis plus 5 left parenthesis y minus 3 right parenthesis right square bracket equals 0 15 x plus 15 minus 4 y plus 12 minus z plus 1 minus 6 z plus 6 plus 2 x plus 2 plus 5 y minus 15 equals 0 17 x plus y minus 7 z plus 21 equals 0$

Exemple 2

Com trobar punts d'un pla?

Trobar un punt d'un pla és molt senzill: es donen valors qualssevol a dues variables i es calcula l'altravariable. Exemple:

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

Agafem, per exemple

$x equals 0 comma space y equals 0 space space space space rightwards double arrow space space z equals negative 5 over 3 space space space space space punt space open parentheses 0 comma 0 comma negative 5 over 3 close parentheses straight o space colon space straight x equals 0 comma space straight z equals 0 space space space rightwards double arrow space space straight y equals negative 5 space space space space space space space punt space left parenthesis 0 comma negative 5 comma 0 right parenthesis$

Propietat intel·lectual i drets d'autoria

GEOMETRIA AFÍ

Equacions d'un pla