Un plano, p, queda determinado cuando se conocen un punto, A, y dos vectores   u→   y  v→ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLw BLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc 9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaalaG aaeiiaiaabccacaqG5bGaaeiiaiaabccaceWG2bGbaSaaaaa@3B93@   paralelos al mismo (vectores de dirección o vectores directores).

¿Qué condición habrá de cumplir un punto X del espacio para que pertenezca al plano p?

El punto X estará sobre el plano cuando, y solo cuando, el vector   AX → MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzY bItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpepe KkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGbbGaamiwaaGaay51Gaaaaa@3949@   sea combinación lineal de   u→   y  v→ MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLw BLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc 9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaalaG aaeiiaiaabccacaqG5bGaaeiiaiaabccaceWG2bGbaSaaaaa@3B93@ . Así pues, habrán de existir números reales l y m tales que:

AX → =λ u→ +μv → MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhi ov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0 xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8HaaeaacaWGbbGaamiwaaGaay 51GaGaeyypa0Jaeq4UdWMabmyDayaalaGaey4kaSIaeqiVd0MabmODayaalaaaaa@40B4@

Fijado un sistema de referencia con origen en O se tiene:

OX→ = OA→ + AX → x→ =a → + AX → MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9w DYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepa e9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWhcaqaaiaad+eacaWGybaacaGLxdcacqGH9 aqpdaWhcaqaaiaad+eacaWGbbaacaGLxdcacqGHRaWkdaWhcaqaaiaadgeacaWGybaacaGLxdcaaeaaceWG4bGbaSaacqGH9aqpceWGHbGb aSaacqGHRaWkdaWhcaqaaiaadgeacaWGybaacaGLxdcaaaaa@4931@

y, en definitiva:

Establecido el sistema de referencia es posible sustituir los vectores de la ecuación por sus coordenadas:

( x, y,  z )=( a 1 , a2 , a3 ) +λ( u1 , u2 ,  u3 ) +μ( v1 , v2 ,  v3 ) ( x,  y, z )=( a1 +λu1 + μv1 ,  a2 +λu 2 +μv2 , a3 + λu3 +μ v3 ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBL nhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqa qpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqadaqaaiaad IhacaGGSaGaaGjbVlaadMhacaGGSaGaaGjbVlaadQhaaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaadggadaWgaaWcbaGaaGymaaqabaGccaGGS aGaaGjbVlaadggadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaGjbVlaadggadaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaacqGHRaWkcqaH7 oaBdaqadaqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGjbVlaadwhadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaGjbVlaadwhadaWga aWcbaGaaG4maaqabaaakiaawIcacaGLPaaacqGHRaWkcqaH8oqBdaqadaqaaiaadAhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaaGjbVlaadAhad aWgaaWcbaGaaGOmaaqabaGccaGGSaGaaGjbVlaadAhadaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaaaeaadaqadaqaaiaadIhacaGGSaGaa GjbVlaadMhacaGGSaGaaGjbVlaadQhaaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaadggadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaH7oaBc aWG1bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqiVd0MaamODamaaBaaaleaacaaIXaaabeaakiaacYcacaaMe8UaamyyamaaBaaaleaacaaIY aaabeaakiabgUcaRiabeU7aSjaadwhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcqaH8oqBcaWG2bWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaay sW7caWGHbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaeq4UdWMaamyDamaaBaaaleaacaaIZaaabeaakiabgUcaRiabeY7aTjaadAhadaWgaaWcb aGaaG4maaqabaaakiaawIcacaGLPaaaaaaa@98E9@

Resultando:

ECUACIONES PARAMÉTRICAS DEL PLANO r: { x= a 1+λ u 1+μ v 1 y= a 2+λ u 2+μ v 2 z= a 3+λ u 3+μ v 3 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2y d9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0 Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr =xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiaacQdadaGabaqaauaabaqadeaaaeaacaWG4bGae yypa0JaamyyamaaBaaaleaacaaIXaaabeaakiabgUcaRiabeU7aSjaadwhadaWgaaWcbaGaaGymaaqabaGccqGHRaWk cqaH8oqBcaWG2bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamyEaiabg2da9iaadggadaWgaaWcbaGaaGOmaaqabaGccqG HRaWkcqaH7oaBcaWG1bWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaeqiVd0MaamODamaaBaaaleaacaaIYaaabeaaaO qaaiaadQhacqGH9aqpcaWGHbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaeq4UdWMaamyDamaaBaaaleaacaaIZaaab eaakiabgUcaRiabeY7aTjaadAhadaWgaaWcbaGaaG4maaqabaaaaaGccaGL7baaaaa@5F78@

Si en el sistema anterior se toman l y m como incógnitas:

{ u 1 λ+ v 1μ =x− a 1 u 2λ+ v 2 μ=y− a 2 u 3λ+ v 3μ=z− a 3 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2Caerbu LwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9L q=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaa eaafaqaaeWabaaabaGaamyDamaaBaaaleaacaaIXaaabeaakiabeU7aSjabgUcaRiaadAhadaWgaaWcbaGaaGymaaqabaGccqaH8oqBcqGH9 aqpcaWG4bGaeyOeI0IaamyyamaaBaaaleaacaaIXaaabeaaaOqaaiaadwhadaWgaaWcbaGaaGOmaaqabaGccqaH7oaBcqGHRaWkcaWG2bWaa SbaaSqaaiaaikdaaeqaaOGaeqiVd0Maeyypa0JaamyEaiabgkHiTiaadggadaWgaaWcbaGaaGOmaaqabaaakeaacaWG1bWaaSbaaSqaaiaai odaaeqaaOGaeq4UdWMaey4kaSIaamODamaaBaaaleaacaaIZaaabeaakiabeY7aTjabg2da9iaadQhacqGHsislcaWGHbWaaSbaaSqaaiaai odaaeqaaaaaaOGaay5Eaaaaaa@5DE4@

Para que el sistema, de tres ecuaciones con dos incógnitas, tenga solución, el rango de la matriz ampliada ha de ser dos, y su determinante cero:

| u1 v 1 x− a1 u2 v 2 y− a2 u3 v 3 z− a3 |=0MathType@MTEF@5 @5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDha rqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firpep eKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaafaqabeWadaaabaGaamyDamaaBaaaleaacaaIXaaabe aaaOqaaiaadAhadaWgaaWcbaGaaGymaaqabaaakeaacaWG4bGaeyOeI0IaamyyamaaBaaaleaacaaIXaaabeaaaOqaaiaadwhadaWgaaW cbaGaaGOmaaqabaaakeaacaWG2bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamyEaiabgkHiTiaadggadaWgaaWcbaGaaGOmaaqabaaakeaa caWG1bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamODamaaBaaaleaacaaIZaaabeaaaOqaaiaadQhacqGHsislcaWGHbWaaSbaaSqaaiaai odaaeqaaaaaaOGaay5bSlaawIa7aiabg2da9iaaicdaaaa@51BF@     

Al desarrollar el determinante resulta:

ECUACIÓN GENERAL O IMPLÍCITA DEL PLANO Ax+By +Cz+D= 0MathType@MTEF@5@5@+=feaaguart1ev2aaatCv AUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rN CHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0Firp epeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadIhacqGHRaWkcaWGcbGaamyEaia bgUcaRiaadoeacaWG6bGaey4kaSIaamiraiabg2da9iaaicdaaaa@4070@

Hay, por último, otra forma de la ecuación del plano denominada segmentaria que resulta muy interesante de cara a la representación gráfica del plano.

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Pincha y arrasta sobre el punto X y se irán mostrando los distintos valores de los parámetros λ y μ.

Creado con GeoGebra