18 02 Spring 2009 Problem Set 1 Due Thursday February 12 at 1 00 PM in 2 106 18 02 Notes Exercices and Solutions are for sale at the Copy Technology Center in the basement of Building 11 This is where to find the exercices labelled 1A 1B etc Problem Sets have two parts A and B Part A has problems from the text with answers to many in the back of the text and problems from the Notes with solutions at the end of the Notes Look at the solutions if you get stuck but try to do as much as possible without them Part A will be graded quickly checking that the problems are there and the solutions not merely copied Part B consists of unsolved problems is worth more points and will be graded more carefully Many of these problems are longer multi part exercises posed here because they do not fit conveniently into an exam or short answer format Homework Rules Collaboration on problem sets is encouraged but 1 Attempt each part of each problem yourself Read each portion of the problem before asking for help If you dont understand what is being asked ask for help interpreting the problem and then make an honest attempt to solve it 2 Write up each problem independently On both Part A and B exercises you are expected to write the answer in your own words 3 Write on your problem set whom you consulted and the sources you used If you fail to do so you may be charged with plagiarism and subject to serious penalties 4 It is illegal to consult materials from previous semesters Reading The material for this problem set is covered in sections D and M of the 18 02 Notes Exercises and Solutions and sections 17 3 18 1 18 2 18 3 18 4 of Simmons Calculus with Analytic Geometry 2nd Edition Part A 10 points Exercises from the Supplementary Notes and Exercises 20 problems 1 2 point each Vectors 1A 1A 1 1A 2 1A 3 1A 5 Dot Product 1B 1B 1 1B 2 1B 4 1B 7 1 Determinants 1C 1C 1 1C 2 1C 3 Cross Product 1D 1D 1 1D 2 1D 3 Equations of Lines and Planes 1E 1E 1 Matrix Algebra 1F 1F 3 1F 4 Solving Square systems Inverse Matrices 1G 1G 1 Cramer s Rule Theorems about Square Systems 1H 1H 3 1H 5 Part B 25 points There are 5 problems worth 5 points each 1 5 points Label the four vertices of a parallelogram in counterclockwise order as OP QR Prove that the line segment from O to the midpoint of P Q intersects the diagonal P R in a point X that is 1 3 of the way from P OP and B OR and express everything in terms to R Hint Let A and B of A 2 5 points The Cauchy Schwarz inequality a Prove from the geometric definition of the dot product the following inequality for vectors in the plane or in space B A B A 1 Under what circumstances does equality hold b If the vectors are plane vectors what does this inequality say in terms and B along the and vectors If the vectors of the components of A are space vectors what does it say in terms of the components of A and B along the vectors k c Give a different proof of this inequality by considering for fixed A and B the function tB A tB f t A 2 Hint First use the algebraic properties of dot product to write f as a quadratic polynomial in t Show that f t is never negative Thus f t can have at most one zero The quadratic formula implies that the coefficients of a quadratic polynomial with at most one zero satisfy a certain inequality 3 5 points Using vector methods show that the general formula for the distance from a point x0 y0 z0 to the plane defined by the equation is ax by cz d 3 ax0 by0 cz0 d a2 b2 c2 4 2 4 5 points Consider the matrix A cos sin sin cos 5 a What happens when a vector is multiplied by A that is what is the geometric relationship between a plane vector v and A v Draw a picture showing the two cases v and v when 3 b Consider A the same matrix as 5 but with in place of Compute the matrix multiplication A A algebraically Deduce immediately without further calculation a formula for the inverse matrix A 1 What is a geometric justification for your conclusion c Use trigonometric identities and matrix multiplication to show that A 1 A 2 A 1 2 What is a geometric justification for this identity 5 5 points Suppose that A is a matrix with determinant zero A 0 and that x1 is a solution of A x b Show that any other solution of this system may be written as x x1 x0 where x0 is a solution of the system A x 0 Apply this principle in the particular case of the system 1 1 x 5 2 2 y 10 to find a general formula for all solutions of this system 6 3 6
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