Question

Use Lagrange multipliers to find the shortest distance between the line (2,4,4)+s(4,1,5) and (1,-3,2)+t(-2,3,1).

Answer 1

to be specific, I don't know how to use the parametric equation of the lines, I understand that I need to take the distance squared (denote f), and the two lines as the constraints(denote g1, g2; I think, though probably not in parametric form?) and the solve f=ag1+bg2 (a, b are the lagrange multipliers), which ends up with the addition of scalars and vectors...

Answer 2

are those points the endpoints of the lines, varying with s & t?

Answer 3

i believe a constraint may be that the vector of the two points forming the least distance between the two lines, will be perpendicular to both lines:
http://mathforum.org/library/drmath/view/51980.html

Answer 4

what about the lagrange multiplier?

Answer 5

and btw, the s and t belong to the real numbers, its just the parametric equation of the lines

Answer 6

I'm getting closer, this page is illuminating, although, it doesn't get into lagrange.
http://www.netcomuk.co.uk/~jenolive/skew.html

Answer 7

let the two points be (2,4,4)+s(4,1,5) and (1,-3,2)+t(-2,3,1)
the line is (2,4,4)+s(4,1,5) - [ (1,-3,2)+t(-2,3,1)]
use distance formula d = sqrt((x2 - x1)^2 + ... )
the two constraints imposed are ... 's' and 't' ... use Lagrange multipliers to minimize the distance function.

Answer 8

honestly, that still doesn't tell me what to do with a function (the distance function which would be squared) that is scalar wise, and the constraints which are vector wise..., maybe if you could show me the exact constraints, and if they are in vector form, please show me how to use it?