I didn't explain very well what I did.
lines
1-2: define Vn as a quaternion vector, and x being a real coefficient.
2-4: define functions f(x) and g(x) that they equal your two lines.
4-6: find Vi such that V2xVi=i, which can be found by premulitiplying by V2^-1
6-10: dilate both f(x) and g(x) by a factor of Vi, then translate both functions. This means that one function is the I axis (xi) and the other is now just a jumble. This is where my stuff falls apart, by multiplying g(x) by Vi, you introduce a real part into the system and g(x)xVi, which makes everything turn to shit.
10-13: What i try to do here is shift the g(x) variant along the real axis (which shouldn't change the line at all, because it's imaginary, but it does because I fucked up.) and remove the i term from the constant. Vs is basically the i intercept of the line.
The distance now, should be the j,k distance away from the i axis. so basically, the distance from the origin to the point (x,y).
This next step I forgot to add:
you can now differentiate the difference, find the turning point, and then substitute the x into the
original line You have dilated the lines, so you need to use the original lines to find the distance. use the i coefficient of h(x) to substitute into f(x). Then find the distance between f(x) and g(x) at the two points found.
question easily done
EDIT: I get your reasoning, both planes will be parallel, even if both vectors are parallel (however, then there will only be two lines not planes). therefore if you find a line perpendicular to the planes, and find the intersections between the lines and the planes, the distance between the two will be the shortest distance between the two lines.
EDIT2: here is a general (working) solution. by substituting 1/c(x+ck) you get an equation of the form below, with no k therm in the constant, and 1 as the coefficient of k in the other term. This does not change the line shape at all.
http://cid-de1e7fef4aaf7e0d.skydrive.live.com/self.aspx/.Public/quarternial%20problem.PNG
Okay, so basically, assign each vector a quaternion value in the pure imaginary realms, then manipulate one vector so it is the i axis, rid yourself of the i value in the other equation's constant then calculate the distance between the two lines. Find the derivative of the distance between the two lines, find the turning point, and then sub into the distance equation and voila, you have the shortest distance between two lines. There's probably a lot of mistakes, but I think the theory is right.
EDIT: I think my theory is wrong. when multiplying by vi, I introduce a real part into the system, and it all goes per shaped, and I completely forget about it.