Linear Algebra problem (Least Squares?)

[Pratha]

Linear Algebra problem (Least Squares? - Distance between lines)

Homework Statement

We have two points R = (x,x,x) and S = (y,3y,-1). All we know is that they are on lines somewhere in 3-space and that they don't cross. Need to find an x and y that minimize || R - S ||²

Homework Equations

A^TAx = A^Tb

The Attempt at a Solution

I tried using the equation above, i.e. inverting (A^TA) and multiplying both sides with that, but the resulting matrix that I got was a 2x1 matrix of zeros. This is definitely not the right answer. I also tried using (C+D(t)-b)²... for each coord and doing a partial derivative for C and D, but I ended up getting the same equation for both derivatives, which I am sure is not right.

I am very confused and not sure where to go from here.

 

[clamtrox]

Why don't you minimize it in the usual way, \nabla ||R-S||^2 = 0 ?

 

[Pratha]

Why don't you minimize it in the usual way, \nabla ||R-S||^2 = 0 ?

That is what I tried. At least that's what I think I tried. That was where the (C+D(t)-b)²... etc, was about in my previous post. (C+Dx - y)² + (C+Dx - 3y)² + (C+Dx + 1)².

But, since the t (x) values are all x's, they cancel with the two's after I do the partial derivative w/respect to D, and both derivatives end up the same. Is there something I'm missing, or am I doing something wrong?