Suppose that x is a C3 path in R3, parametrized by arclength, with κ 0. Suppose that the...

Suppose that x is a C³ path in R3, parametrized by arclength, with κ 0. Suppose that the image of x lies in the xy-plane.

(a) Explain why x must have a constant binormal vector.

(b) Show that the torsion τ must always be zero. Note that there is really nothing special about the image of x lying in the xy-plane, so that this exercise, combined with the results of Exercise 28, shows that the image of x is a plane curve if and only if τ is always zero and if and only if B is a constant vector.