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Conceptual Problems 3 1. Suppose that r,(f) and r2(/) are two curves which intersect at a point P. If P =ri(h) and P = 12(42), does this mean that & = by? If so, briefly explain why. If not, give a counterexample. 2. Suppose that r, () is the equation of a curve and one defines r2(() = r, ()/) for A a positive constant (basically 12 is using a different "time scale" than the one used for ri). How is the velocity vector of r2 related to the one of r,? What about the acceleration vectors? If one defines ra(?) = Fi(l-4), how is the velocity vector of Ta related to the one of r,? What about the acceleration vectors? Make sure you justify your answers. 3. Is the velocity vector v(f) of a curve r() always perpendicular to the acceleration vector a(f)? If so, briefly explain why. If not, give a counterexample. 4. Suppose that r(f) is a vector valued function, v(f) is its velocity, and a() is its acceleration, all functions defined for all f e (-co, co), Assume r(() x v(1) = c for all i E (-co, co), where e is a constant vector. Show that r(t) and a(() are parallel for all & E (-co. co). Give an explicit example of a path r(?) such that r(() = -2a(1)