18.02 Spring 2009 Problem Set 1Due Thursday, February 12 at 1:00 PM in 2-10618.02 Notes, Exercices and Solutions are for sale at the Copy Technol-ogy Center in the basement of Building 11. This is where to find the exerciceslabelled 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 ofthe 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 withoutthem. Part A will be graded quickly, checking that the problems are there andthe solutions not merely copied.Part B consists of unsolved problems, is worth more points, and will begraded more carefully. Many of these problems are longer multi-part exercisesposed here because they do not fit conveniently into an exam or short-answerformat.Homework RulesCollaboration on problem sets is encouraged, but1. Attempt each part of each problem yourself. Read each portion ofthe problem before asking for help. If you dont understand what is beingasked, ask for help interpreting the problem and then make an honestattempt to solve it.2. Write up each problem independently. On both Part A and B ex-ercises you are expected to write the answer in your own words.3. Write on your problem set whom you consulted and the sourcesyou used. If you fail to do so, you may be charged with plagiarism andsubject to serious penalties.4. It is illegal to consult materials from previous semesters.ReadingThe material for this problem set is covered in sections D and M of the 18.02Notes, Exercises and Solutions, and sections 17.3, 18.1, 18.2, 18.3, 18.4 of Sim-mons, Calculus with Analytic Geometry, 2nd Edition.Part A (10 points)Exercises from the Supplementary Notes and Exercises (20 problems, 1/2 pointeach):• 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-5Part B (25 points)There are 5 problems worth 5 points each1. (5 points) Label the four vertices of a parallelogram in counterclockwiseorder as OP QR. Prove that the line segment from O to the midpoint ofP Q intersects the diagonal P R in a point X that is 1/3 of the way from Pto R. (Hint: Let~A = OP and~B = OR, and express everything in termsof~A and~B.)2. (5 points) The Cauchy-Schwarz inequality(a) Prove from the geometric definition of the dot product the followinginequality for vectors in the plane or in space:|~A ·~B| ≤ |~A||~B| (1)Under what circumstances does equality hold?(b) If the vectors are plane vectors, what does this inequality say in termsof the components of~A and~B along the ˆı and ˆ vectors? If the vectorsare space vectors, what does it say in terms of the components of~Aand~B along the vectors ˆı, ˆ,ˆk?(c) Give a different proof of this inequality by considering, for fixed~Aand~B, the functionf(t) = (~A + t~B) · (~A + t~B) (2)(Hint: First, use the algebraic properties of dot product to write fas a quadratic polynomial in t. Show that f (t) is never negative.Thus f(t) can have at most one zero. The quadratic formula impliesthat the coefficients of a quadratic polynomial with at most one zerosatisfy a certain inequality.)3. (5 points) Using vector methods, show that the general formula for thedistance from a point (x0, y0, z0) to the plane defined by the equationax + by + cz = d (3)is|ax0+ by0+ cz0− d|√a2+ b2+ c2(4)24. (5 points) Consider the matrixAθ=cos θ −sin θsin θ cos θ(5)(a) What happens when a vector is multiplied by Aθ, that is, what isthe geometric relationship between a plane vector ~v and Aθ~v? Drawa picture showing the two cases ~v = ˆı and ~v = ˆ when θ = π/3.(b) Consider A−θ, the same matrix as (5) but with −θ in place of θ. Com-pute the matrix multiplication A−θAθalgebraically. Deduce imme-diately (without further calculation) a formula for the inverse matrixA−1θ. What is a geometric justification for your conclusion?(c) Use trigonometric identities and matrix multiplication to show thatAθ1Aθ2= Aθ1+θ2. What is a geometric justification for this identity?5. (5 points) Suppose that A is a matrix with determinant z ero: |A| = 0,and that ~x1is a solution of A~x =~b. Show that any other solution of thissystem may be written as ~x = ~x1+~x0, where ~x0is a solution of the systemA~x =~0.Apply this principle in the particular case of the system1 −12 −2xy=510(6)to find a general formula for all solutions of this system
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