Dubtes freqüents sobre geometria

GEOMETRIA AFÍ

Posició relativa entre dues rectes

Donades dues rectes de l'espai veiem com trobar la seva posició relativa depenent de:

- Si tenim les equacions implícites:

r: $r colon space open curly brackets table row cell A subscript 1 x plus B subscript 1 y plus C subscript 1 z plus D subscript 1 equals 0 end cell row cell A subscript 2 x plus B subscript 2 y plus C subscript 2 z plus D subscript 2 equals 0 end cell end table close space space space space space space space space space space s colon space open curly brackets table row cell A subscript 3 x plus B subscript 3 y plus C subscript 3 z plus D subscript 3 equals 0 end cell row cell A subscript 4 x plus B subscript 4 y plus C subscript 4 z plus D subscript 4 equals 0 end cell end table close space$

considerarem la matriu de coeficients i la matriu ampliada del sistema format per les equacions de les dues rectes:

$M equals open parentheses table row cell A subscript 1 end cell cell B subscript 1 end cell cell C subscript 1 end cell row cell A subscript 2 end cell cell B subscript 2 end cell cell C subscript 2 end cell row cell A subscript 3 end cell cell B subscript 3 end cell cell C subscript 3 end cell row cell A subscript 4 end cell cell B subscript 4 end cell cell C subscript 4 end cell end table close parentheses space space space space space space space space space space space space M apostrophe equals open parentheses table row cell A subscript 1 end cell cell B subscript 1 end cell cell C subscript 1 space space minus D subscript 1 end cell row cell A subscript 2 end cell cell B subscript 2 end cell cell C subscript 2 subscript 1 space space minus D subscript 2 end cell row cell A subscript 3 end cell cell B subscript 3 end cell cell C subscript 3 subscript 1 space space minus D subscript 3 end cell row cell A subscript 4 end cell cell B subscript 4 end cell cell C subscript 4 subscript 1 space space minus D subscript 4 end cell end table close parentheses$

i mirarem els rangs de M i M'

Rectes coincidents rang M = rang M' = 2

Rectes paral·leles rang M = 2, rang M' = 3

Rectes secants rang M = rang M' = 3

Rectes que es creuen: rang M = 3, rang M' = 4

- Si sabem un punt i un vector director de cada recta.

$r e c t a space r colon space p u n t space A comma space v e c t o r space d i r e c t o r space u with rightwards arrow on top space space r e c t a space s colon space p u n t space B comma space v e c t o r space d i r e c t o r space v with rightwards arrow on top space$

estudiarem els vectors $u with rightwards arrow on top comma space v with rightwards arrow on top comma space stack A B with rightwards arrow on top$ i els punts A i B segons els cassos

Rectes coincidents:

$bold u with bold rightwards arrow on top bold parallel to bold v with bold rightwards arrow on top bold space bold space bold space bold space bold italic i bold space bold space bold space bold space bold space bold italic A bold element of bold italic s space space space space left parenthesis l l e g i m colon " u with rightwards arrow on top space é s space p a r a l times l e l space a space v with rightwards arrow on top space i space e l space p u n t space A space é s space d e space l a space r e c t a space s right parenthesis space space space space space$

Rectes paral·leles:

Rectes secants:

$bold u with bold rightwards arrow on top not parallel to bold v with bold rightwards arrow on top bold space bold space bold space bold space bold italic i bold space bold space bold space bold space bold space bold italic d bold italic e bold italic t bold space bold left parenthesis bold u with bold rightwards arrow on top bold comma space bold v with bold rightwards arrow on top bold comma bold space stack bold A bold B with bold rightwards arrow on top bold right parenthesis bold equals bold 0 space space$
(les dues rectes estan contingudes en un pla)

Rectes que es creuen:

$bold u with bold rightwards arrow on top not parallel to bold v with bold rightwards arrow on top bold space bold space bold space bold space bold italic i bold space bold space bold space bold space bold space bold italic d bold italic e bold italic t bold space bold left parenthesis bold u with bold rightwards arrow on top bold comma space bold v with bold rightwards arrow on top bold comma stack bold A bold B with bold rightwards arrow on top bold right parenthesis bold not equal to bold 0 space space$

(les dues rectes no estan contingudes en un pla)