Topic: Is it possible to resolve Alisp something so hard?

[SOFITO_SOFT]

Hello swamp people:
My problem:


 
I have all x y z, which are real numbers (may be 0)
I have p3 and p4 which are points of the form (xyz) with real numbers ...
I know that t is a unit vector of the form (LMN), where (+ L^2  M^2  N^2) = 1
Is it possible to deduce L M N ? I find myself unable ... but maybe some genius of algebra and matrices knows how to do it .... greatly be appreciated  .... Regards.

PD.: Explanation in case one to encourage resolve this gibberish: (LMN) is the unit vector of the axis of the cylinder that contains the points p1 p2 p3 and p4 .... given in a coordinate system such that:
p1 is 0,0,0
p2 would be along the x axis (y = 0 , z = 0)
p3 would have z = 0
p1 p2 p3 would form a plane in which:
p4 would have coordinates (x y z) any ...

With this vector ( L M N )   projected p1 p2 p3 p4 to a plane through 0,0,0 (p1) and has as its normal vector (LMN) and the bari-center of any of the triangles that form with those 4 projections, is the center of the circle "base" of the cylinder, according to various experts who have published in several books:
http://hal.inria.fr/docs/00/07/24/27/PDF/RR-4195.pdf
I think so 

It is very confusing but has its charm ....

[SOFITO_SOFT]

Hello:
Aid that help me:

UCS  W 
coordinates :

e1 X = 10.0000     Y = 20.0000     Z = 30.0000   
e2 X = 68.6121     Y = 69.4890     Z = 55.0000   
p1 X = 19.9659     Y = 28.4147     Z = 34.2508   
p2 X = 19.9659     Y = 15.5262     Z = 33.3698   
p3 X = 62.5897     Y = 51.8786     Z = 46.7476   
p4 X = 59.0755     Y = 51.8786     Z = 56.1075   

VECTO-UNIT  e1->e2 =  (0.726461 0.613386 0.309859)
VECTO-UNIT  e2->e1 =  (-0.726461 -0.613386 -0.309859)

UCS for 3 points p1 p2 p3
coordinates :

e1 X = 18.9098      Y = -8.5562     Z = -7.3712 
e2 X = -36.0300     Y = 50.2656     Z = -1.7879 
p1 X = 0.0000       Y = 0.0000      Z = 0.0000   
p2 X = 20.6840      Y = 0.0000      Z = 0.0000   
p3 X = -16.8449     Y = 43.6901     Z = 0.0000   
p4 X = -22.1523     Y = 38.5687     Z = 6.7499

thanks in advance...
 

[SOFITO_SOFT]

Hi all:
With such data, the equality works!!!
 77294.0  = 77294.0
Greetings.