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1. Rectes en l'espai

Donats

vector director 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 k 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 italic k bold v subscript bold 1 end cell row cell bold y bold equals bold a subscript bold 2 bold plus bold italic k bold v subscript bold 2 end cell row cell bold z bold equals bold a subscript bold 3 bold plus bold italic k bold v subscript bold 3 end cell end table close curly brackets

Equació contínua

               fraction numerator bold x bold minus bold a subscript bold 1 over denominator bold v subscript bold 1 end fraction bold equals fraction numerator bold y bold minus bold a subscript bold 2 over denominator bold v subscript bold 2 end fraction bold equals fraction numerator bold z bold minus bold a subscript bold 3 over denominator bold v subscript bold 3 end fraction

Equació implícita   

              open bold A bold x bold plus bold B bold y bold plus bold C bold z bold plus bold D bold equals bold 0 bold A bold apostrophe bold x bold plus bold B bold apostrophe bold y bold plus bold C bold apostrophe bold z bold plus bold D bold apostrophe bold equals bold 0 close curly brackets


Exemple

Equacions de la recta que passa pel punt A equals left parenthesis negative 1 comma 3 comma 1 right parenthesis i té vector director 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 k 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 2 k end cell row cell straight y equals 3 minus k end cell row cell straight z equals 1 plus 5 k end cell end table close curly brackets

            

Equació contínua:

   fraction numerator x plus 1 over denominator 2 end fraction equals fraction numerator y minus 3 over denominator negative 1 end fraction equals fraction numerator z minus 1 over denominator 5 end fraction

Equació implícita

  l'obtenim de l'anterior:

   open table attributes columnalign right end attributes row cell negative 1 left parenthesis x plus 1 right parenthesis equals 2 left parenthesis y minus 3 right parenthesis end cell row cell 5 left parenthesis x plus 1 right parenthesis equals 2 left parenthesis z minus 1 right parenthesis end cell end table close curly brackets space space space space rightwards double arrow space space open table attributes columnalign right end attributes row cell negative x minus 1 equals 2 y minus 6 end cell row cell 5 x plus 5 equals 2 z minus 2 end cell end table close curly brackets space space space space space rightwards double arrow space space open table attributes columnalign right end attributes row cell x plus 2 y minus 5 equals 0 end cell row cell 5 x minus 2 z plus 7 equals 0 end cell end table close curly brackets   

Observació: si tenim l'equació implícita d'una recta, per saber punt i vector director veieu:   

Punt i vector director d'una recta donada amb equació implícita.

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