Question 3. Consider a vector function r(t). 3.1. Show that if r(f) 1 7 for all r. then r'(t) 1 # for all t. [2 points] 3.2. Let # = (a, b, c) be a vector. Show that if "'(t) 1 7 for all t, then r() lies in some plane perpendicular to v. Hint: Consider the derivative of red in order to show that there is some constant d such that if (x. v. =) = r(t) for some f, then ar + by + et = d. [3 points] Question 4. In class we showed that if f (f) is a vector function that lies in a sphere {{ s.t. (of = r), then f(t) I f'(t) for all t. Now we will show the converse: if f (t) I f'(t) for all t, then f lies in a sphere. 4.1. Express (f . f)(t) in terms of (f . S')(!). [1 point] 4.2. Assume f(t) 1 S'(t). Show that there is some real number r such that (/ (t)| =r for all t. Hint: Show that the function (f . A)(t) = [/()| is constant by considering its derivative. Since / ()) is always nonnegative, it will follow that [/(t)| = \\(f . f)(?) is also constant. (3 points]