Distance between two parametric lines?

Posted on 2015-10-13 06:11:00 in Math by MICHELE

Suppose the line L1 is parametrized by r1(t) = <2t+5,3t,t-3>, and the line L2 is parametrized by r2(t)=. Define the function f(s,t) to be the square of the distance between r1(t) and rs(s)

a) Think about what happens to f(s,t) as s,t->+-infinity. How many critical points ought f have.
b) write the formula for f(s,t) and all its critical points . Identify f as one of the "quadratic surfaces"
c) find s* and t* so that the distance between r1(t*) and r2(s*) is as small as possible. the line segment between these points has an interesting geometric property related to L1 and L2. What is the property?

How am I supposed to think about what happens to f(s,t) as s,t->+-infinity without finding out the function first?

I found the distance between the lines to be the vector v=r2(s)-r1(s) and i defined f(s,t) as v dot v
and I got f(s,t)= -5s^2+35t^2-14s-10t+2st+11 how am I supposed to know if it is an ellipsoid, hyperboloid, paraboloid etc

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