Use the following scheme to prove that in E3 a straight line is the shortest distance between two points. Let X : [a,b] → E3 be a curve, and set x(a) = p, x(b) = q.
Hi everyone - I wanted to make sure I was on the right track. I believe I need to show that the distance between p and q is less than or equal to the arclength of some L - thereby showing that the shortest distance is always the straight line between them. Here's what I have so far:
d(p,q) = q - p
X(t) = p + t(q - p) = p(1 - t) + tq (for t:[0,1])
X'(t) = q - p
L = q - p
Thereby proving that L = d(p,q).
But I feel like that was too easy. Did I calculate the arclength right? Thanks very much for any help!