Calculation of the intersection of two 3D lines in space.

The proposed solution is for a point-vector pair. I'm looking for the equations for two lines, each defined by two points in space.

Line 1:  (x1,y1,z1) and (x2,y2,z2)
Line 2: (x3,y3,z3) and (x4,y4,z4)

For each of your line segments, you can convert to point-vector form by using either point, and the difference between the two points as the direction vector.

       (x ,y ,z ) =  (x1,y1,z1)  =  t*( x2-x1), (y2-y1), (z2-z1))   for the first line.    t=0  gives  (x1,y1,z1)     and    t=1  gives  (x2,y2,z2)

Sorry for getting back to this so late.
In the example provided, the final answer is unclear to me (i.e. the (X,Y,Z) location of the intersection point).
Is the final intersection point (X,Y,Z) = (5,6,2)?

Also as an aside, I'm not getting the cross product he is getting in his example (see below)

from article
"Using Bensegueni's method, we find

  a(V1 x V2) = a(-10,-11,-13)"

I get  a(V x V2) =  a (-10, 11, -13)   (i.e. middle term positive).

In 2D, if two lines aren't parallel, it exists, for sure, an interception point.
In 3D there is another requirement: besides to be not parallel, the lines must be coplanar, say, they must be in a same plane.

The John Taylor's article Intersecting Lines in 3D: describes these conditions and gives some examples.

Jose

Assuming the lines are defined by their end points (x1, y1, z1) and (x2, y2, z2) lets use the general parametric equation:

      x = x1 + (x2 - x1)*t
      y = y1 + (y2 - y1)*t
      z = z1 + (z2 - z1)*t

if lines are
Line A --> (xa1,ya1, za1,  xa2,ya2, za2) and
Line B --> (xb1,yb1, zb1,  xb2,yb2, zb2),

first define the parametric equations

Line A
x = xa1 + (xa2 - xa1) * ta
y = ya1 + (ya2 - ya1) * ta
z = za1 + (za2 - za1) * ta

Line B
x = xb1 + (xb2 - xb1) * tb
y = yb1 + (yb2 - yb1) * tb
z = zb1 + (zb2 - zb1) * tb

To check if they are parallel,  get the vectors for each line.

Line A
(the components of the vector are the coefficients of ta):
M1 = (xa2 - xa1) * i + (ya2 - ya1) * j + (za2 - za1) * k

Line B
(do the same as for Line A, by using the coefficients of tb):
M2 = (xb2 - xb1) * i + ...

If M1 and M2 are equal or proportional, the lines are parallel, so there is no interception point.
If not, go ahead.

Now set the x and y values.
Of course, x and y must be the same in both lines equations, so,

x = xa1 + (xa2 - xa1) * ta = xb1 + (xb2 - xb1) * tb
and
y = ya1 + (ya2 - ya1) * ta = yb1 + (yb2 - yb1) * tb

By solving tha above equations, we have values for ta and tb.
Now, just apply these values in the z equations to chek is they are true.
If so, you found the interception point, if not, they don't intercept each another.

Jose