MATH 21A Final: Math 21a Final Exam 1 Fall 2009
Document Summary
Answers which are illegible for the grader cannot be given credit: show your work. Except for problems 1-3, we need to see details of your computation: no notes, books, calculators, computers, or other electronic aids can be allowed, you have 180 minutes time to complete your work. There are two unit vectors ~v, ~w for which the sum ~v + ~w has length 1/3. For any three vectors, we have |(~u ~v) ~w| = |(~v ~w) ~u|. Denote by d(p, l) the distance from a point p to a line l in space. P and any two lines l, k in space, we have d(p, l) + d(p, k) d(l, k). For any three vectors ~u, ~v, ~w, the relation |~u (~v ~w)| |~u||~v||~w| holds. Assume a curve ~r(t) has constant speed 1 and constant curvature 1 everywhere. Then the curve ~r(2t) has constant speed 2 and constant curvature 1/2 every- where.
