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Berkeley COMPSCI 184 - Assignment 2

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The purpose of this assignment is to test your prerequisite math background. In general, you should find this assignment to be fairly easy. This assignment must be done solo. To submit, please slide un-der Prof. O’Brien’s door at 527 Soda Hall.Introduction: Some of these questions may look a little difficult. If they do, consider reviewing how dot products and cross products are formed, how matrices are multiplied with vectors and with each other, how determinants are formed and what eigenvalues are. This exercise could take a very long time if you don’t think carefully about each problem, but good solutions will use one or two lines per question. Many textbooks supply expressions for the area of a triangle and for the volume of a pyramid, which you may find very useful. You must write neatly. If the assignment cannot be read, it cannot be graded. If you find that you make a mess as you work out the solution, then you should use scrap paper for working it out and copy it down neatly for turning in. If you are feeling particularly intrepid, consider formatting your solutions with LaTeX. If you are feeling particularly perverse, consider formatting your solutions using Microsoft’ Word’s equation editor.I know that many of you have access to solutions to this, or a very similar assignment from previous years. To be honest, if you need to cheat on this assignment then you should just go drop the class now.CS 184: Foundations of Computer Graphics page 1 of 3Fall 2011Prof. James O’BrienAssignment #2Point Value: 15 pointsDue Date: Sept. 6th, 1:00 pmUNIVERSITY OF CALIFORNIACollege of Engineering, Department of Electrical Engineering and Computer SciencesComputer Science Division: Fall 2006 CS-184 Foundations of Computer GraphicsProfessor: James O’BrienAssignment 1Point value: 40 ptsThis homework must be done individually. Submission date is Friday, September 8th 2006, at10:00am under the door of 633 Soda HallIntroduction: Some of these questions may look a little di⇥cult. If they do, consider review-ing how dot products and cros s products are formed, how matrices are multiplied with vectors andwith each other, how determinants are formed and what eigenvalues are. This exercise could takea very long time if you don’t think carefully about each problem, but good solutions will use aboutone line per question. Many textbooks supply expressions for the area of a triangle and for thevolume of a pyramid, which you may find very useful.You must write neatly. If the assignment cannot be read, it cannot be graded. Ifyou find that you m ake a mess as you work out the solution, then you should use scrap paperfor working it out and copy it down neatly for turning in. If you are feeling particularly intrepid,consider formatting your solutions with LATEX.I know that many of you have access to solutions to this, or a very similar assignment fromprevious years. To be honest, if you need to cheat on this assignment then just go drop the classnow.Question 1: Two vectors in the plane, i and j, have the following properties (i · i means thedot product between i and i): i · i = 1, i · j = 0, j · j = 1.1. Is there a vector k, that is not equal to i, such that: k · k = 1, k · j = 0? What is it? Arethere many vectors with these properties?2. Is there a vector k such that: k · k = 1, k · j = 0, k · i = 0? Why not?3. If i and j were vectors in 3D, how would the answers to the above questions change?Question 2: For three p oints on the plane (x1, y1), (x2, y2) and (x3, y3) show that the deter-minant of⇧⇤x1y11x2y21x3y31⇥⌃⌅is proportional to the area of the triangle whose corners are the three points. If these points lie ona straight line, what is the value of the determinant? Does this give a useful test to tell whetherthree points lie on a line? Why do you think so?Question 3: The equation of a line in the plane is ax + by + c = 0. Given two points on theplane, show how to find the values of a, b, c for the line that passes through those two points. Youmay find the answer to question 2 useful here.Question 4: Let e1= (1, 0, 0), e2= (0, 1, 0), e3= (0, 0, 1). Show that if {i, j, k} is {1, 2, 3},{2, 3, 1}, or {3, 1, 2}, then ei⇤ ej= ek, where ⇤ is the cross product. Now show that if {i, j, k} is{1, 3, 2}, {3, 2, 1} or {2, 1, 3}, then ei⇤ ej= ek.UNIVERSITY OF CALIFORNIACollege of Engineering, Department of Electrical Engineering and Computer SciencesComputer Science Division: Fall 2006 CS-184 Foundations of Computer GraphicsProfessor: James O’BrienAssignment 1Point value: 40 ptsThis homework must be done individually. Submission date is Friday, September 8th 2006, at10:00am under th e door of 633 Soda HallIntroduction: Some of these questions may look a little di⇥cult. If they do, consider review-ing how dot products and cross products are formed, how matrices are multiplied with vectors andwith each other, how determinants are formed and what eigenvalues are. This exercise could takea very long time if you don’t think carefully about each problem, but good solutions will use aboutone line per question. Many textbooks supply expressions for the area of a triangle and for thevolume of a pyramid, which you may find very useful.You must write neatly. If the assignment cannot b e read, it cannot be graded. Ifyou find that you make a mess as you work out the solution, then you should use scrap paperfor working it out and c opy it down neatly for turning in. If you are feeling particularly intrepid,consider formatting your solutions with LATEX.I know that many of you have acc es s to solutions to this, or a very similar assignment fromprevious years. To be honest, if you need to cheat on this assignment then just go drop the classnow.Question 1: Two vectors in the plane, i and j, have the following properties (i · i means thedot product b etween i and i): i · i = 1, i · j = 0, j · j = 1.1. Is there a vector k, that is not equal to i, such that: k · k = 1, k · j = 0? What is it? Arethere many vectors with these properties?2. Is there a vector k such that: k · k = 1, k · j = 0, k · i = 0? Why not?3. If i and j were vectors in 3D, how would the answers to the above questions change?Question 2: For three points on the plane ( x1, y1), (x2, y2) and (x3, y3) show that the deter-minant of⇧⇤x1y11x2y21x3y31⇥⌃⌅is proportional to the area of the triangle whose corners are the three points. If these points lie ona


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Berkeley COMPSCI 184 - Assignment 2

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