# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

4 Pages

### HW#3

Course: MATHEMATIC MA116, Fall 2011
School: Stevens Institute of...
Rating:

Word Count: 1122

#### Document Preview

116 MA Calculus II HW 3 (Solutions) Due: March 4, 2011 1. Find the following by directly applying the definition of cross product, i.e., right-hand rule: j a) i = k j d) k = i b) k = i j e) i = 0 jj ( ( ( ) jj c) i i = 2. Let v = x, y and v = ) ) f) k = 0 jj x cos + y sin , x sin + y cos . Show the angle between v and v is . v v Using the definition of dot product...

Register Now

#### Unformatted Document Excerpt

Coursehero

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
116 MA Calculus II HW 3 (Solutions) Due: March 4, 2011 1. Find the following by directly applying the definition of cross product, i.e., right-hand rule: j a) i = k j d) k = i b) k = i j e) i = 0 jj ( ( ( ) jj c) i i = 2. Let v = x, y and v = ) ) f) k = 0 jj x cos + y sin , x sin + y cos . Show the angle between v and v is . v v Using the definition of dot product we have cos = v v (1) v v = x, y x cos + y sin , x sin + y cos = (2) = x 2 + y 2 v (3) v= ( x cos + y sin ) 2 + ( x cos + y sin )= 2 (x 2 cos + xy sin ) + ( xy sin + y 2 cos = ) x 2 ( cos 2 + sin 2 ) + y 2 ( cos 2 + sin 2 = ) (x 2 + y 2 ) cos x2 + y 2 Putting (1), (2), and (3) into our definition of dot product we get cos = cos . Since the dot product is only defined for 0 , cos is a one to one function for this given domain. Hence, = . 3. Show that if u and v are any vectors, then the vectors v u + u v and v u u v are orthogonal. For two vectors to be orthogonal their dot product must be equal to zero. So we take the dot product of the two vectors keeping in mind properties of dot product on page 651. ( v u + u v ) ( v u u v ) = v u v u + v u ( u v ) + u v v u + u v ( u v ) 2 2 = v u u v u u v + u v v u u v v 2 2 2 2 = v u u v =0 4. Consider the two planes 2x + 3y 4z = 9 3x 4 y + 2 z = 11 a) Find the angle between them. b) Find the parametric and symmetric equations for the line of intersection of the two planes. c) At what point does the line from part b) intersect the y z plane? (a) The angel between the two planes is the acute angel between the normals of the two planes. The normals for both planes are n1 2,3, 4 and n2 3, 4, 2 . To get the angle between these two = = vectors we use definition of dot product. cos = n1 n2 14 cos 1 14 = = 29 n1 n2 29 ( ) x0 y0 z0 (b) The parametric equations of a line are x = + at , y = + bt , z = + ct , where ( x0 , y0 , z0 ) is a point on the line and v = a, b, c is a vector parallel to the line. To find a point on the line of intersection we pick a value for x, y, or z and plug it into both equations of the planes. Ill pick x = 1. Plugging in x = 1 we get 3y 4z = 11 4 y + 2 z = 8 Solving for these two we equations get y = z = have ( x0 , y0 , z0= (1, 1, 2 ) . Keep 1 and 2. So we ) in mind your point will probably be different then mine. Now, to find the parallel vector v since the line of intersection is located on both planes we realize vector v will be orthogonal to both normals of the plane. Hence, k i j v = n = 3 4 = 10i 16 17 k . n 2 j 1 2 3 4 2 Giving us the following parametric equations: x = 1 10t y =1 16t z= 2 17t Again, your parametric equations might look different. To get the symmetric equations we solve for t. = t x 1 y +1 z 2 == 10 16 17 (c) For a line to intersect the y z plane, x = 0. Plug x = 0 into the symmetric equations and we get y = 3 5 and z = 37 10 . 5. Consider the line that is parallel to the z axis and goes through the point ( 0, 2,0 ) . Find the equation of the plane that contains this line and the point ( 2,0,0 ) . To find the equation of a plane we need a point on the plane and a normal vector to the plane. We are given two points on the plane and can use either. We need the normal vector, so we select two vectors in the plane and form their cross product. One vector in the plane is given, i.e., the direction of the z axis. Ill use k . For the other vector, Ill use the vector from the given point on the line to the other given point. That is 2, 2,0 . Then, n = 2, 2,0 0,0,1 ( ) j = 2i 2 k () j = 2 2i j =2i 2 So we now write the equation of the plane using either point. 2 ( x 2 ) 2( y 0) = 0 2x 2 y = 4 x+ y= 2 6. Velocity in a Magnetic Field The force F on a proton moving at velocity v m/s in a uniform magnetic field B (in teslas) is = q v B in newtons, where = 1.6 1019 coulombs. Calculate F if B = 0.0004k teslas and v has F q ( ) magnitude 106 m/s in the direction + k . [Feel free to use the properties of cross product on the bottom j of page 656 to make the calculation easier.] v To find the vector v recall that the unit vector u in the direction of v is given by u = . So v = v u . v The vector + k has length 2 , and since v has magnitude 106 , j +k j v = 106 2 Therefore, the force (in newtons) is +k j 400q + k k = 400q i F = v B = 6 q q 10 j 0.0004k = 2 2 2 ( = ( ) 400 (1.6 1019 ) 2 ) i ( 4.5 1017 ) i (( ))
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Stevens Institute of Technology - MATHEMATIC - MA116
Ma 116 - Calculus IIHw-4 (Solutions)Due: March 25, 2011Pledge and Sign:An important component of these written exercises is the quality of your presentation,including: legibility, organization of the solution, and clearly stated reasoning whereappro
Stevens Institute of Technology - MATHEMATIC - MA116
Ma 116 - Calculus IIHw-5 (Solutions)Due: April 8, 2011Pledge and Sign:An important component of these written exercises is the quality of your presentation,including: legibility, organization of the solution, and clearly stated reasoning whereapprop
Stevens Institute of Technology - MATHEMATIC - MA116
Ma 116 - Calculus IIHw-6 (Solutions)Due: Apr 25, 2011Pledge and Sign:An important component of these written exercises is the quality of your presentation,including: legibility, organization of the solution, and clearly stated reasoning whereappropr
Abu Dhabi University - FIN - 123
International Business MGT 440Section 01 Quiz # 1Time: 15 MinutesPoints: 20Attempt all questions. Mark the correct answer.1. The deepening relationship and broadening interdependence among people from.different parts of the world is known as@ glob
UBC - CPSC - 344
Name: Signature:Student ID Number:CPSC 344 2005-06 Term 1 Midterm Exam 50 minutesDepartment of Computer Science University of British Columbia K. MacLean Exam Instructions (read carefully): 1. Immediately sign the first page of the exam with your signa
UBC - CPSC - 344
Name: Signature:Student ID Number:CPSC 344 2005-06 Term 1 Midterm Exam 50 minutes SOLUTIONS + commentsDepartment of Computer Science University of British Columbia K. MacLean Exam Instructions (read carefully): 1. Immediately sign the first page of the
UBC - CPSC - 344
Name:Student ID Number:Signature:CPSC 344 2006-07 (T1) Final Exam[PARTIAL] SOLUTIONSDepartment of Computer ScienceUniversity of British ColumbiaK. MacLeanExam Instructions (read carefully):1. Immediately sign the first page of the exam withyour
UBC - CPSC - 344
ame: Signature:Student ID umber:CPSC 344, 2007-08 Winter Term 1 Midterm Exam 90 minutesDepartment of Computer Science University of British Columbia S. Wolfman Exam Instructions (read carefully): 1. Sign this page in the space provided to indicate your
UBC - CPSC - 344
ame: Sample Soln (yours may differ!)Student ID umber:Signature:CPSC 344, 2007-08 Winter Term 1 Midterm Exam90 minutesDepartment of Computer ScienceUniversity of British ColumbiaS. WolfmanExam Instructions (read carefully):1. Sign this page in the
UBC - CPSC - 344
Name:Student ID Number:Signature:CPSC 344 2008W1 Midterm Exam90 minutesDepartment of Computer ScienceUniversity of British ColumbiaK. MacLeanExam Instructions (read carefully):1. Sign this page in the space provided to indicate youragreement wit
UBC - CPSC - 344
Name:Student ID Number:Signature:CPSC 344 2008W1 Midterm Exam 90 minutes SOLUTIONS + commentsDepartment of Computer Science University of British Columbia K. MacLean Exam Instructions (read carefully): 1. Sign this page in the space provided to indica
UBC - CPSC - 344
Name: Signature:Student ID Number:CPSC 444 2002-03 (T1) Final ExamDepartment of Computer Science University of British Columbia K. MacLean Exam Instructions (read carefully): 1. Sign the first page of the exam with your signature in the space provided
UBC - CPSC - 344
CPSC 320 Sample Midterm 1February 2009Name:Signature:Student ID: You have 50 minutes to write the 5 questions on this examination.A total of 45 marks are available. Justify all of your answers. You are allowed to bring in one hand-written, double-
UBC - CPSC - 344
CPSC 320 Sample Midterm 1February 2009[12] 1. Answer each of the questions with either true or false. You must justify each of youranswers; an answer without a justication will be worth at most 1.5 out of 4.[4] a. If we can use the Master theorem to d
UBC - CPSC - 344
CPSC 320 Sample Midterm 2March 2009Name:Signature:Student ID: You have 50 minutes to write the 4 questions on this examination.A total of 40 marks are available. Justify all of your answers. You are allowed to bring in one hand-written, double-sid
UBC - CPSC - 344
CPSC 320 Sample Midterm 2March 2009[12] 1. Skip Lists[4] a. When we augment a skip list to support efcient Order-Statistics queries, we storea count value with each pointer of the skip list. What does count represent, andhow is it used by SkipListSel
UBC - CPSC - 344
CPSC 320 Sample Final ExaminationApril 2009Name:Signature:Student ID: You have 2.5 hours to write the 8 questions on this examination. A total of 110 marks are available. Justify all of your answers, except if the questionsays not to.Question No
UBC - CPSC - 344
CPSC 320 Sample Final ExaminationApril 2009[10] 1. Answer each of the following questions with true or false. Give a short justicationfor each of your answers.[5] a. 6n O(5n )This is false:66nlim n = limn 5n + 5= +Solution :nwhich means that
UBC - CPSC - 344
CPSC 320 Midterm 1Monday, October 18th, 2010[13] 1. Short answers[4] a. We frequently prove an O(n2 ) bound on the worst-case running time T (n) of analgorithm, without proving any bound on T (n). However, we would never provean (n2 ) bound on T (n)
UBC - CPSC - 344
CPSC 320 Midterm 2Monday, November 15th, 2010[16] 1. Short Answers[4] a. Your boss asks you to design a divide-and-conquer algorithm to solve a problemwhose input is an array of measurements that contain temperature data. What are thetwo main issues
UBC - CPSC - 344
CPSC 320 Sample Final ExaminationDecember 2010Name:Signature:Student ID: You have 2.5 hours to write the 8 questions on this examination. A total of 100 marks are available. Justify all of your answers, except if the questionsays not to.Question
UBC - CPSC - 344
CPSC 320 Sample Final ExaminationDecember 2010[10] 1. Answer each of the following questions with true or false. Give a short justicationfor each of your answers.[5] a. 6n O(5n )This is false:6n6lim n = limn 5n + 5= +Solution :nwhich means t
UBC - CPSC - 344
CPSC 320 Sample Midterm 1October 2010Name:Signature:Student ID: You have 50 minutes to write the 4 questions on this examination.A total of 40 marks are available. Justify all of your answers. You are allowed to bring in one hand-written, double-s
UBC - CPSC - 344
CPSC 320 Sample Midterm 1October 2010[12] 1. Answer each of the questions with either true or false. You must justify each of youranswers; an answer without a justication will be worth at most 1.5 out of 4.[4] a. 3n+2 + 5 O(3n1 ).Solution : This is t
UBC - CPSC - 344
CPSC 320 Sample Midterm 2November 2010Name:Signature:Student ID: You have 50 minutes to write the 5 questions on this examination.A total of 40 marks are available. Justify all of your answers. You are allowed to bring in one hand-written, double-
UBC - CPSC - 344
CPSC 320 Sample Midterm 2November 2010[6] 1. Short Answers[3] a. If we can not use a recursion tree to prove a tight bound on the value of a function T (n) dened by a recurrence relation, can we use the Master theorem instead?Explain why or why not.S
UBC - CPSC - 344
CPSC 313, 05w Term 1 Final ExamDate: December 12, 2005; Instructor: Mike FeeleyThis is a closed book exam; no notes; you may use calculators to perform simple arithmetic calculations.Pages of potentially useful notes appear at the end of the exam. Answ
UBC - CPSC - 344
CPSC 313, 05w Term 1 Midterm 1Date: October 7, 2005; Instructor: Mike FeeleyThis is a closed book exam; no notes; you may use calculators to perform simple arithmetic calculations.Answer in the space provided; use the backs of pages if needed. There ar
UBC - CPSC - 344
CPSC 313, 05w Term 1 Midterm 1 SolutionsDate: October 7, 2005; Instructor: Mike Feeley1. (10 marks) Short answers.1a. What is the advantage of using two different registers (i.e., %ebp and %esp) to store virtualaddresses to the runtime stack?The stac
UBC - CPSC - 344
CPSC 313, 05w Term 1 Midterm 2Date: November 9, 2005; Instructor: Mike FeeleyThis is a closed book exam; no notes; you may use calculators to perform simple arithmetic calculations.NOTE: the last page shows the PIPE architecture diagram. Answer in the
UBC - CPSC - 344
CPSC 313, 05w Term 1 Midterm 2 SolutionsDate: November 9, 2005; Instructor: Mike Feeley1. (10 marks) Short answers.1a. Briey explain how an n-channel MOSFET transistor closes the circuit between its source anddrain when its gate voltage is high.The h
UBC - CPSC - 344
CPSC 313, 06w Term 1 Midterm 1Date: October 4, 2006; Instructor: Norm HutchinsonThis is a closed book exam; no notes; you may use calculators only if you dont trust your own brain toperform simple arithmetic calculations. Answer in the space provided;
UBC - CPSC - 344
CPSC 313, 06w Term 1 Midterm 1 SolutionsDate: October 4, 2006; Instructor: Norm Hutchinson1. (8 marks)1a.Short answers.Is the address of a local variable in a C function determined statically or dynamically?Briey explain.(2 marks)It is determined
UBC - CPSC - 344
CPSC 313, 06w Term 1 Midterm 2Date: November 17, 2006; Instructor: Norm HutchinsonThis is a closed book exam; no notes; you may use a calculator if you wish. Answer in the space provided;use the backs of pages if needed. There are 6 questions on 8 page
UBC - CPSC - 344
CPSC 313, 06w Term 1 Midterm 2 SolutionsDate: November 17, 2006; Instructor: Norm Hutchinson1. (10 marks)Short answers.1a.(2 marks) In the standard pipeline (F, D, E, M, W), does stalling stage D one cycle cause stalls orbubbles for any other cycles
UBC - CPSC - 344
CPSC 313, 06w Term 2 Midterm 1Date: February 9, 2007; Instructor: Norm HutchinsonThis is a closed book exam; no notes; no calculators. Answer in the space provided; use the backs of pagesif needed.There are 6 questions on 4 pages, totaling 42 marks.Y
UBC - CPSC - 344
CPSC 313, 06w Term 2 Midterm 1 SolutionsDate: February 9, 2007; Instructor: Norm Hutchinson1. (8 marks)Short answers.1a.(2 marks) Is the address of a global variable in a C program determined statically (before theprogram runs) or dynamically (while
UBC - CPSC - 344
CPSC 313, 06w Term 2 Midterm 2Date: March 23, 2007; Instructor: Norm HutchinsonThis is a closed book exam; no notes; you may use a calculator if you wish. Answer in the space provided;use the backs of pages if needed.There are 6 questions on 8 pages,
UBC - CPSC - 344
CPSC 313, 06w Term 2 Midterm 2 SolutionsDate: March 23, 2007; Instructor: Norm Hutchinson1. (12 marks)Short answers.1a. (2 marks) Describe the difference between stalling and creating a bubble.Stalling means that an instruction remains in a particula
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #1Due: Sunday, January 21, 2006 at 11: 59 PM (thirteen-hour grace period)Instructions: Hand all of your solutions in on paper.1. On an IA32 machine (e.g., lin01.ugrad.cs.ubc.ca), using gcc, compile the following C progra
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #1 Solution1. I edited the output slightly to make it shorter..bss_counter:.space 4.text_swap:pushl%ebpmovl%esp, %ebpsubl\$8, %espmovl%ebx, (%esp)movl8(%ebp), %edxmovl16(%ebp), %ebxmovl%esi, 4(%esp)movl
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #2Due: Sunday, January 28, 2006 at 11:59 PM (thirteen-hour grace period)All of your solutions should be turned in on paper.1. Variables declared in a C program can be stored in either registers or memory, and if inmemor
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #2 Solution1. The only illegal operation is taking the address of i, since it is in a register. Note the similarity of allof the accesses to the variables that are stored in memory: a, l, and g. Despite the dierences in h
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #3Due: Sunday, February 4, 2007 at 11:59 PM (thirteen-hour grace period)1. Write in C a function int gcd(int x, int y); that returns the greatest common divisorof its two arguments. You may assume that they are both posi
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #3 - Solution1. (a) int gcd(int a, int b)cfw_if (a = b)return a;else if (a &gt; b)return gcd(a - b, b);elsereturn gcd(a, b - a);.file&quot;gcdrec.c&quot;.text.p2align 4,15.globl gcd.typegcd, @functiongcd:pushl%ebpmov
UBC - CPSC - 344
CS 313, Winter 2006 - Term 2Assignment 4: Defusing a Binary BombAssigned: February 5, Due: Sunday, February 18, 11:59:59 PM1IntroductionThe nefarious Dr. Evil has planted a slew of binary bombs on our machines. A binary bomb is a programthat consist
UBC - CPSC - 344
CS 313, Winter 2006 - Term 2Assignment 5: HCL and y86Assigned: February 26, Due: Sunday, March 4, 11:59PM (with theusual 13 hour grace period)Instructions: Hand in all solutions on paper.1. Write an HCL expression for a boolean signal implies, true w
UBC - CPSC - 344
CPSC 313 06W Term 2 Problem Set #5 - Solution 1. bool implies = !a | b; 2. The four signals divide into 2 classes, sort0 and sort3 choose the minimum or maximum elements, which look like this. sort3 is identical to sort0, with the comparisons changed from
UBC - CPSC - 344
CPSC 313, Winter 2006 - Term 2Understanding the Y86 Architectureand its Sequential ImplementationAssigned: March 2, Due: Sunday, March 11, 11:59PMFor this problem set, you may either work by yourself or in a group of two. If you choose to work in agr
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #6 - Solution1. iaddl:StageFetchDecodeExecuteMemoryWrite backPC updateiaddl V, rBicode:ifun M1 [PC]rA:rB M1 [PC + 1]valC M4 [PC + 2]valP PC + 6valB R[rB]valE valB + valCsetCCR[rB] valEPC valP2. leave:Sta
UBC - CPSC - 344
CPSC 313, Winter 2006 - Term 2PipeliningAssigned: March 9, Due: Sunday, March 18, 11:59PMAll of these questions are to be handed in on paper.1. Consider a ve-stage pipeline with stage gate delays of 150 ps, 75 ps, 100 ps, 100 ps and 175 ps anda memor
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #7 - Solution1.(a)1185ps= 5.41 109 cycles per second or 5.41 GHz(b) 5.41 Gops (instructions per second)(c) 5 * 185 = 925 ps per instruction(d)1i. 600ps = 1.66 109 cycles per second or 1.66 GHzii. 1.66 Gops (instr
UBC - CPSC - 344
CPSC 313, Winter 2006 - Term 2Problem Set #8Assigned: March 26, Due: Sunday, April 1, 11:59PMYou will hand in all of your solutions to this problem set on paper.1. What is SRAM? Where do you typically nd it? What is good about it? What is bad about it
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #8 - Solution1. SRAM stands for Static Random Access Memory. It can be typically found in caches, on or off thecore. It is fast and persistent, but it requires more transistors so is more expensive per unit of memorythat
UBC - CPSC - 344
CPSC 313, Winter 2006 - Term 2Problem Set #9Assigned: April 2Due Wednesday, April 11, 13:00 pm (no grace period)1. Textbook 10.112. Textbook 10.123. Textbook 10.134. Consider a virtual-memory system with the following parameters PTBR = physical ad
UBC - CPSC - 344
CPSC 31306W Term 2Problem Set #9 - Solution1.A. 00 0010 0111 1100B.VPNTLBITLBTTLB hit?page fault?PPN0x90x10x2NN0x17C. 0101 1111 1100D.COCICTcache hit?cache byte?2.0x00xf0x17N-A. 00 0011 1010 1001B.VPNTLBITLBTTLB hit?pa
American University in Bulgaria - MANAGEMENT - 101
Tutorials: Lecture 1 &amp; 3Introduction to Logistics &amp;Transportationby: Granit BerishaSeptember 29,2011Logistics is that part of SCM that plans, implements, and controlstheefficient, effective, forward and reverse flow and storage of goods,services,
University of Phoenix - HISTORY - HIS/135
Running head: THE CIVIL RIGHTS MOVEMENT1The Civil Rights MovementPamela NewsomeHIS/135September 8, 2011Rebecca HowzeTHE CIVIL RIGHTS MOVEMENT2The Civil Rights MovementThe one event that I consider to be the most influential to the success of the
McGill - PSYC - 215
READING NOTES: CHAPTER 5PERSUASION-Some examples are described They talk about Germans, Nazis, the Holocaust, propaganda, etc. One of theseexamples is particularly related to us as Canadian citizens:o Surveys shortly before the in Iraq revealed that C
Ball State - ASTRO - 101
SUBJECT: HIST 150The West in the WorldASSIGNMENT TEN: An Era of Totalitarianism and World War II.READING ASSIGNMENT:Textbook: Marvin Perry, Western Civilization, A Brief History, 6th ed., BostonHoughton Mifflin Company, 2008, chapters *19 and 20, pp.4
Ball State - ASTRO - 101
SUBJECT: HIST 150The West in the WorldASSIGNMENT ELEVEN: The West in a Global Age and the Epilogue.READING ASSIGNMENT:Textbook: Marvin Perry, Western Civilization, A Brief History, 6th ed., Boston:Houghton Mifflin Company, 2008, chapter 21 and Epilogu