Suppose that x is a C3 path in R3 with torsion τ always equal to 0.
(a) Explain why x must have a constant binormal vector (i.e., one whose direction must remain fixed for all t).
(b) Supposewe have chosen coordinates so that x(0) = 0 and that v(0) and a(0) lie in the xy-plane (i.e., have no k-component). Then what must the binormal vector B be?
(c) Using the coordinate assumptions in part (b), show that x(t) must lie in the xy-plane for all t. (Hint: Begin by explainingwhy v(t) · k = a(t) · k = 0 for all t. Then show that if
x(t) = x(t)i + y(t)j + z(t)k,
we must have z(t) = 0 for all t.)
(d) Now explain how we may conclude that curves with zero torsion must lie in a plane.
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