Verified answer

P(-4,-5,9): Q(5-1,-3); R(-3,1,5) Vectores: QR=(9,4,-12) PR=(1,6,-4)

(x,y,z)=P1 + t(PR) + s(PR)

(x,y,z)=(-4,-5,9) + t(9,4,-12) + s(1,6,-4)

x=-4 + 9t + s_>_|x+4__y+5__z-9__x+4__y+5|

y=-5 + 4t + 6s>_|__9 ___4___-12___9____4_| =0

z=9 -12t -4s__>_|__1____6___-4____1____6_|

-16x-64-12y-60+54z-486 - (4z-36-72x-288-36y-180)

72x-16 +36y-12y -4z+54z +504-610=0

Exacto 56X+24Y+50Z-106=0 es la ecuación implícita del plano a partir de tres puntos.

verifiquemos

Supongamos

a x + b y + c z = 1

-4 a - 5 b + 9 c = 1

5 a - 1 b - 3 c = 1

-3 a+ 1 b + 5 c = 1

D =

((-4)(-1)(5)+(5)(1)(9)+(-3)(-5)(-3)) - ((9)(-1)(-3)+(-3)(1)(-4)+(5)(-5)(5))

= (20+45-45)-(27+12-125)

= 20 - (-86)

= 106

D * a =

((1)(-1)(5)+(1)(1)(9)+(1)(-5)(-3)) - ((9)(-1)(1)+(-3)(1)(1)+(5)(-5)(1))

= (-5+9+15)-(-9-3-25)

= 19 - (-37)

= 56

a = 28/53

****************

D * b =

((-4)(1)(5)+(5)(1)(9)+(-3)(1)(-3)) - ((9)(1)(-3)+(-3)(1)(-4)+(5)(1)(5))

= (-20+45+9)-(-27+12+25)

= 34 - (10)

= 24

b = 12/53

***************

D * c =

((-4)(-1)(1)+(5)(1)(1)+(-3)(-5)(1)) - ((1)(-1)(-3)+(1)(1)(-4)+(1)(-5)(5))

= (4+5+15)-(3-4-25)

= 24 - (-26)

= 50

c = 25/53

queda

(28/53) x + (12/53) y + (25/53) z = 1

ó

28 x + 12 y + 25 z = 53

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Concidimos salvo un factor de 2

Saludos