Minimizing Distance between Lines in R3 | Squared Distance and Equations

[derryck1234]

Homework Statement

In R³, consider the line l given by the equations {x=t,y=t,z=t} and the line m given by the equations {x=s,y=2s-1,z=1}. Let P be a point on l, and let Q be a point on m. Find the values of t and s that minimize the distance between the lines by minimizing the squared distance abs(P-Q).

Homework Equations

[P] = A(A^TA)^-1A^T

The Attempt at a Solution

Let the basis for l be span{(1, 1, 1)} and the basis for m be span{(1,2,0),(0,-1,1)}

From here on I actually don't know what to do:( Do I have to apply the formula to both lines?

Please help!

 

[tiny-tim]

hi derryck1234!

what's the difficulty?

P is (t,t,t) and Q is (s,2s-1,1), so minimise PQ² (as the question says )

 

[derryck1234]

Ok. But are you saying that I should use calculus to do that? Because in my textbook, whenever we need to use calculus, the question has a line before it saying: FOR THOSE READERS WHO HAVE STUDIED CALCULUS...

This one doesn't?

To be honest, I don't even think I remember how to do it...would it entail working out abs(P-Q), then finding the derivative and then setting it to zero?

Thanks

Derryck

 

[tiny-tim]

Hi Derryck!

Ok. But are you saying that I should use calculus to do that?

no, you could https://www.physicsforums.com/library.php?do=view_item&itemid=107" instead

 

[derryck1234]

Thanks tiny tim. I do hope my correspondence maths course goes ok. It is not easy let me tell you...doin maths via correspondence:( Especially in South Africa!