# Unidad 3. Espacios Vectoriales Básicos 3.4 Producto escalar, vectorial y triple producto escalar Triple Producto Escalar

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Dados tres vectores de R 3 u = (a, b, c) v = (r, s, t) w = (x, y, z) con ellos se busca el de formar primero el producto vectorial y luego el producto punto de u con v × w. Hay una forma sencilla de obtener el resultado de este triple producto

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❯♥✐❞❛❞✸✳❊♣❛❝✐♦❱❡❝♦✐❛❧❡❇✐❝♦  ✸✳✹♦❞✉❝♦❡❝❛❧❛✱✈❡❝♦✐❛❧②✐♣❧❡♣♦❞✉❝♦❡❝❛❧❛ ❚✐♣❧❡♦❞✉❝♦❊❝❛❧❛ ❉❛❞♦❡✈❡❝♦❡❞❡  R 3 u  = ( a,b,c )  v  = ( r,s,t )  w  = ( x,y,z ) ❝♦♥❡❧❧♦❡❜✉❝❛❡❧❞❡❢♦♠❛♣✐♠❡♦❡❧♣♦❞✉❝♦✈❡❝♦✐❛❧②❧✉❡❣♦❡❧♣♦❞✉❝♦♣✉♥♦❞❡  u ❝♦♥   v  × w ✳❍❛②✉♥❛❢♦♠❛❡♥❝✐❧❧❛❞❡♦❜❡♥❡❡❧❡✉❧❛❞♦❞❡❡❡✐♣❧❡♣♦❞✉❝♦❡❝❛❧❛✱❝♦♠♦❧♦♠✉❡❛❡❧❝❧❝✉❧♦ ✐❣✉✐❡♥❡✿ u · v × w  =  a  s ty z  − b  r tx z  +  c  r sx y  =  a b cr s tx y z  = [ u,v,w ] ▲❛❧✐♠❛✐❣✉❛❧❞❛❞❞❡❄♥❡✉♥❛♥♦❛❝✐♥♣❛❛❡❧✐♣❧❡♣♦❞✉❝♦❡❝❛❧❛✉❡❡✈✐❛❡❝✐❜✐♦❞♦❡❧❞❡❡♠✐✲ ♥❛♥❡✳ ♦♣✐❡❞❛❞✶  ❊❝❝❧✐❝♦✱❡❞❡❝✐✱♣✉❡❞❡♦♠❛❡❝♦♠♦♣✐♠❡❢❛❝♦♦♦✈❡❝♦❝♦♥❛❧❞❡✉❡♥♦❡ ❛❧❡❡❡❧♦❞❡♥❝❝❧✐❝♦✭❝♦❧♦✉❡❧♦❡✈❡❝♦❡❡♥✉♥❛❝✐❝✉♥❢❡❡♥❝✐❛❝✉②♦❡❝♦✐❞♦♣❛❡♣✐♠❡♦ ♣♦  u ✱❧✉❡❣♦♣♦   v ②❄♥❛❧♠❡♥❡♣♦   w ✮❀❡♥♠❜♦❧♦  [ u,v,w ] = [ v,w,u ] = [ w,u,v ] ❉❡♠♦❛❝✐♥✳  ❊♥❡❡❝❛♦  [ u,v,w ] =  a b cr s tx y z  = 2 csx  +  cry  +  btx  + 2 brz  + 2 aty  +  asz [ v,w,u ] =  r s tx y za b c  = 2 csx  +  cry  +  btx  + 2 brz  + 2 aty  +  asz [ w,u,v ] =  x y za b cr s t  = 2 csx  +  cry  +  btx  + 2 brz  + 2 aty  +  asz ♦♣✐❡❞❛❞✷  ❈❛♠❜✐❛❞❡✐❣♥♦❛❧✐♥❡❝❛♠❜✐❛❞♦❞❡❧♦✈❡❝♦❡✭♥♦❡✉❡❡♦❝❛♠❜✐❛❡❧♦❞❡♥❝❝❧✐❝♦✮❀♣♦❡❥❡♠♣❧♦✱ [ u,v,w ] =  − [ v,u,w ] ❊♥❡❡❝❛♦  − [ v,u,w ] =  − v · u × w  =  v · w × u  = [ v,w,u ] = [ u,v,w ] ♦♣✐❡❞❛❞✸  ❙❡❞✐✐❜✉②❡♦❜❡❧❛✉♠❛✱❡❞❡❝✐✱ [ u 1  +  u 2 ,v,w ] = [ u 1 ,v,w ] + [ u 2 ,v,w ] ❋❛❝✉❧❛❞❞❡❈✐❡♥❝✐❛❯◆❆▼ ●❡♦♠❡❛❆♥❛❧✐❝❛■ ♦❢✳❊❡❜❛♥❘✉❜♥❍✉❛❞♦❈✉③  ✶   ❯♥✐❞❛❞✸✳❊♣❛❝✐♦❱❡❝♦✐❛❧❡❇✐❝♦  ✸✳✹♦❞✉❝♦❡❝❛❧❛✱✈❡❝♦✐❛❧②✐♣❧❡♣♦❞✉❝♦❡❝❛❧❛  ❉❡♠♦❛❝✐♥✳  ❊♥❡❡❝❛♦✐  u 1  = ( u 1 1 ,u 1 2 ,u 1 3 ) ②   u 2  = ( u 2 1 ,u 2 2 ,u 2 3 ) ✳❊♥♦♥❝❡  [ u 1  +  u 2 ,v,w ] = ( u 1  +  u 2 ) · v × w  u 1 1  +  u 2 1  u 1 2  +  u 2 2  u 1 3  +  u 2 3 r s tx y z  =  u 1 1  u 1 2  u 1 3 r s tx y z  +  u 2 1  u 2 2  u 2 3 r s tx y z  =  u 1  · v × w  +  u 2  · v × w = [ u 1 ,v,w ] + [ u 2 ,v,w ] ♦♣✐❡❞❛❞✹  ❙❛❝❛❡❝❛❧❛❡✱❡♦❡✱ [ λu,v,w ] =  λ [ u,v,w ] ❉❡♠♦❛❝✐♥✳  ❊♥❡❡❝❛♦✐  u 1  = ( u 1 1 ,u 1 2 ,u 1 3 ) ②   λ  ∈ R ✳❊♥♦♥❝❡  [ λu 1 ,v,w ] = ( λu 1 ) · v × w  λu 1 1  λu 1 2  λu 1 3 r s tx y z  =  λ  u 1 1  u 1 2  u 1 3 r s tx y z  =  λ ( u 1  · v × w )=  λ [ u 1 ,v,w ] ♦♣✐❡❞❛❞✺  ❊①♣❡❛❡❧✈♦❧✉♠❡♥♦✐❡♥❛❞♦❞❡❧♣❛❛❧❡❧❡♣♣❡❞♦✭  u,v,w ✮❞❡❧❛❋✐❣✉❛ ❊❛♣♦♣✐❡❞❛❞❡❞❡♠✉❡❛♦♠❛♥❞♦❡♥❝✉❡♥❛❧❛✐♥❡♣❡❛❝✐♥❣❡♦♠✐❝❛❞❡❧♦♣♦❞✉❝♦ ❛♥❡✐♦❡✿ [ u,v,w ] =  u · v × w  =   u   v × w  cos  φ ❞♦♥❞❡  φ  = ∠  ( u,v × w ) ❋❛❝✉❧❛❞❞❡❈✐❡♥❝✐❛❯◆❆▼ ●❡♦♠❡❛❆♥❛❧✐❝❛■ ♦❢✳❊❡❜❛♥❘✉❜♥❍✉❛❞♦❈✉③  ✷   ❯♥✐❞❛❞✸✳❊♣❛❝✐♦❱❡❝♦✐❛❧❡❇✐❝♦  ✸✳✹♦❞✉❝♦❡❝❛❧❛✱✈❡❝♦✐❛❧②✐♣❧❡♣♦❞✉❝♦❡❝❛❧❛ ②✱❝♦♠♦  v × w ❡♣❡♣❡♥❞✐❝✉❧❛❛   v ②   w ✱❡❧♥♠❡♦    u  cos  φ ❡❧❛❛❧✉❛❞❡   P  ( u,v,w ) ❡♣❡❝♦❞❡ ❧❛❜❛❡❢♦♠❛❞❛♣♦❡❧♣❛❛❧❡❧♦❣❛♠♦❞❡❡♠✐♥❛❞♦♣♦  u ②   w ✱❝✉②❛❡❛❡♣❡❝✐❛♠❡♥❡    u,v × w  ❋❛❝✉❧❛❞❞❡❈✐❡♥❝✐❛❯◆❆▼ ●❡♦♠❡❛❆♥❛❧✐❝❛■ ♦❢✳❊❡❜❛♥❘✉❜♥❍✉❛❞♦❈✉③  ✸
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