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

specifically, I don't know how to work with the equations of the lines in that form.

Answer 2

I know I will use the distance squared, with the constraints as those two lines, but the equation of the lines is not in a form I can work with...

Answer 3

What is a form that you can work with? (2,4,4)+s(4,1,5) means the line joining the points (2,4,4) and (6,5,9). ( (6,5,9) = (2,4,4)+ (4,1,5) ).

Answer 4

What I mean is that the lines are not in the form like ax+by=cz, I don't know how to work with the parametric equations of the lines

Answer 5

u have the following sets
\[ \large L_1=\{(2,4,4)+s(4,1,5):s\in\mathbb{R}\} \]
and
\[ \large L_2=\{(1,-3,2)+t(-2,3,1):t\in\mathbb{R}\} \]
then the distance between any couple of points of these sets would be
\[ \large\sqrt{(4s+2t+1)^2+(s-3t+7)^2+(5s-t+2)^2}=d(s,t) \]
this is the function u have to minimize.

Answer 6

thanks :D

Answer 7

u r welcome