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3. Posició relativa entre recta i pla

Donats la recta  i el pla d'equacions:

  r colon open curly brackets table attributes columnalign left end attributes row cell A subscript 1 x plus B subscript 1 y plus C subscript 1 z plus D subscript 1 equals 0 end cell row cell A subscript 1 x plus B subscript 1 y plus C subscript 1 z plus D subscript 1 equals 0 end cell end table close space space space space space space space capital pi colon space A x plus B y plus C z plus D equals 0

per estudiar la seva posició relativa o podem veure de dues maneres:

a) Considerant  les matrius

      M equals open parentheses table row cell A subscript 1 end cell cell B subscript 1 end cell cell C subscript 1 end cell row cell A subscript 2 end cell cell B subscript 2 end cell cell C subscript 2 end cell row A B C end table close parentheses space space space space space space space space space space space space space space M apostrophe equals open parentheses table row cell A subscript 1 end cell cell B subscript 1 end cell cell C subscript 1 space space minus D subscript 1 end cell row cell A subscript 2 end cell cell B subscript 2 end cell cell C subscript 2 space space minus D subscript 2 end cell row A B cell C space space space space minus D end cell end table close parentheses

Llavors: 

rang M rang M'
Recta continguda en el pla 2 2
Recta i pla paral·lels 2 3
Recta i pla secants 3 3


Exemple:

   


b) A partir del vector director i un punt de la recta y el vector normal del pla:

       recta r:   vector director u with rightwards arrow on top, punt A

       pla Π:      vector normal n with rightwards arrow on top

Recta continguda en el pla:          

      bold u with bold rightwards arrow on top bold times bold n with bold rightwards arrow on top bold equals bold 0 bold space bold space bold space bold space bold space bold italic i bold space bold space bold space bold space bold italic A bold element of bold pi

             

Recta i pla paral·lels:                 

  bold u with bold rightwards arrow on top bold times bold n with bold rightwards arrow on top bold equals bold 0 bold space bold space bold space bold space bold space bold italic i bold space bold space bold space bold space bold italic A bold not an element of bold pi


Recta i pla secants:          

  bold u with bold rightwards arrow on top bold times bold n with bold rightwards arrow on top bold not equal to bold 0 bold space


Exemple 

   


Com a cas particular de recta i pla secants es pot considerar el cas de recta i pla perpendiculars.  La recta i el pla seran perpendiculars quan el vector u with rightwards arrow on topdirector de la recta r , i el vector i el vector normal n with rightwards arrow on toppla Π siguin paral·lels. O sigui: 

                                     v with rightwards arrow on top equals k n with rightwards arrow on top

   O expressat en components:

                u with rightwards arrow on top equals left parenthesis u subscript 1 comma u subscript 2 comma u subscript 3 right parenthesis n with rightwards arrow on top equals left parenthesis n subscript 1 comma n subscript 2 comma n subscript 3 right parenthesis                 u subscript 1 over n subscript 1 equals fraction numerator begin display style u subscript 2 end style over denominator begin display style n subscript 2 end style end fraction equals fraction numerator begin display style u subscript 3 end style over denominator begin display style n subscript 3 end style end fraction


Exemple 

   


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