Prove that a(s) is a straight line if and only if its tangent lines are all parallel.
Frenet serret theorem
The Attempt at a Solution
I'm confused on the direction "if the tangent lines are parallel then a(s) is a straight line".
Assume all the tangent lines of a(s) are parallel. So the tangent vector T is the same for all points xo on the curve a(s) and the values of T(s) of any two points on the curve are parallel. Thus T(s) is constant, and T'(s)=0 which implies that the curvature is zero, and thus a(s) must be a straight line.