7. Let r: R → R" be a path parameterized by arclength - i.e. the speed is a constant one. a) Prove that r (s) I7"(s) for all s. b) Let T(s) = r'(s) and N T"(s) |F"(s)|| Define a new vector field B(s)...

Extracted text: 7. Let r: R → R" be a path parameterized by arclength - i.e. the speed is a constant one. a) Prove that r (s) I7"(s) for all s. b) Let T(s) = r'(s) and N T"(s) |F"(s)|| Define a new vector field B(s) = T(s) × N(s). Prove %3D that ||B(s)|| = 1. Note: T, N, and B form a mutually perpendicular trihedron of unit vectors along r called the Frenet frame. This frame is crucial to the study of the geometry of curves and is used in the study of objects moving in space.