Homework 2. Chapters 1, 2, 3. Vector addition, dot products and cross products
|
|
- Teresa Jennings
- 5 years ago
- Views:
Transcription
1 Homework 2. Chapters 1, 2, 3. Vector addition, dot products and cross products 2.1 Dynamics project description Submit a short (handwritten or typed) paragraph describing your physical system. Include (hand-drawn) schematics showing relevant points, particles, bodies, constraints, etc. Include an interesting question that you would like to answer. 2.2 Sine and cosine review. Trigonometry plays a central role in kinematics, particularly in the formation of rotation matrices. Referring to the figure below, express b1 x, b1 y, b2 x,andb2 y in terms of sin(θ) andcos(θ). b2 x 1 θ b2 y 1 θ b1 y b1 x = b1 y = b2 x = b2 y = b1 x 2.3 Right-handed, orthogonal, unitary basis. Draw a right-handed orthogonal (mutually perpendicular) basis consisting of the unit vectors a x, a y, a z. 2.4 Perpendicular vectors. The vectors v 1 = x a x +2a y +3a z and v 2 =4a x +5a y +6a z are expressed in terms of orthogonal unit vectors a x, a y, a z. Find the value of x so v 1 and v 2 are perpendicular. x = 2.5 Column matrices and vectors. 1 The column matrix 2 is identical to the vector a x +2a y +3a z. True/False Vector concepts: Addition The following vector and column matrix addition produce equivalent results. True/False. Note: a x, a y, a z and b x, b y, b z are sets of orthogonal unit vectors. Explain: a x +2a y +3a z + 4b x +5b y +6b z = = Copyright c by Paul Mitiguy 317 Homework 2
2 2.7 Calculating vector dot products with bases. The figure to the right shows a right-handed (dextral) set of orthogonal unit vectors,, n z. The vectors u, v, w are defined as: u = n z v = x + y + z n z w = n z n z (a) Use the distributive law for dot products to write u v in terms of,, n z, etc. u v = 2x + 2y + 2z n z (b) Use the definition of the dot product to calculate,, etc. = = n z = = = n z = n z = n z = n z n z = (c) In view of your previous two results, calculate u v. u v = (d) As shown in Section , the dot product u v is relatively easy to calculate when,, n z are orthogonal unit vectors. When two arbitrary vectors a and b are expressed in terms of orthogonal unit vectors as shown below, the dot product a b can be calculated as a = a x + a y + a z n z b = b x + b y + b z n z a b = a x b x + a y b y + a z b z In view of this short-cut, calculate u v, u w, and v w. u v =2x +3y +4z u w = v w = (e) MG : Modify and submit the following MotionGenesis file to calculate u w and v w. Note: The following commands show the calculation of u v. % File: CalculateDotProductsWithBasis.al % RigidFrame N Variable x, y, z u> = 2*Nx> + 3*Ny> + 4*Nz> v> = x*nx> + y*ny> + z*nz> w> = 5*Nx> - 6*Ny> + 7*Nz> udotv = Dot( u>, v> ) Save CalculateDotProductsWithBasis.all Quit n z 2.8 Getting started with MotionGenesis. Go to and click on Getting Started. Follow the directions up to Vector Operations. Printoutandsubmit your firstdemo.al file with your homework. Continue through Vector Operations and also submit your vectordemo.al. Copyright c by Paul Mitiguy 318 Homework 2
3 2.9 Definition of a dot product and its use for calculating angles. The figure to the right shows a rectangular parallelepiped (block) of sides 2, 3, and 4. Unit vectors,, n z are directed along the sides of the block as shown. The points A, B, C and D are located at corners of the block. (a) Express r C/A (C s position vector from A) in terms of,, n z. r C/A = (b) Find a numerical value for r C/A r C/A. Next, use equation (2.3) to calculate the magnitude of r C/A (the distance from A to C). r C/A r C/A = r C/A = (c) Using equation (2.1), calculate the unit vector u directed from A to C in terms of,, n z. Next, find the unit vector v directed from A to D in terms of,, n z. u = 3 2 v = 13 (d) Calculate BAC, the angle between line AB and line AC. Next, calculate CAD, the angle between line AC and line AD. BAC = CAD = (e) MG : Modify and submit the following MotionGenesis file to calculate r D/A, v, and CAD. % File: DotProductsToCalculateAngles.al % RigidFrame N Point A, B, C, D B.SetPosition( A, -2*Ny> ) C.SetPosition( B, 3*Nx> ) distancefromatoc = GetMagnitude( C.GetPosition(A) ) u> = C.GetPosition(A) / distancefromatoc anglebacrad = GetAngleBetweenVectors( B.GetPosition(A), C.GetPosition(A) ) anglebacdeg = anglebacrad * ConvertUnits( radians, degrees ) Save DotProductsToCalculateAngles.all Quit Copyright c by Paul Mitiguy 319 Homework 2
4 2.10 Dot products and distance calculations. The figure to the right shows a crane whose cab A supports a boom B that swings a wrecking ball C o. To prevent the wrecking ball from accidently destroying nearby cars, the distance between the nearest car, point N o, and the tip of the boom, point BC, must be controlled. N x A L B θ B B θ C L C C o (a) Express the position vector of BC from N o in terms of x, L B, and the unit vectors,andb x. r BC/No = + (b) Using the distributive property for dot-multiplication of vectors, i.e., (a + b) (c + d) = a c + a d + b c + b d express r BC/No r BC/No in terms of x, L B,and b x. r BC/No r BC/No = (c) Using the definition of the dot-product in equation (2.2), calculate b x. b x = (d) Using your previous two results, rewrite r BC/No r BC/No in terms of x, L B,andθ B. r BC/No r BC/No = (e) Using equation (2.3) to calculate the magnitude of r BC/No, express the distance from N o to BC in terms of x, L B,andθ B, and calculate its value whe=20, L B =10, and θ B =30. r BC/N o = = 29.1 (f) Two colleagues are confused by your use of mixed-bases vectors (i.e., r BC/No = x + L B b x ), and ask you to verify the position vector of B from N o canbeexpressedintheuniform-basis as shown below. Use this uniform-basis expression to verify your previous result for r BC/N o. Note: This uniform-basis approach necessitates the simplifying trigonometric identity sin 2 (θ B )+ cos 2 (θ B )=1. r BC/No = [x + L B cos(θ B )] + L B sin(θ B ) (g) Optional : Calculate the distance from N o to C o in terms of x, L B, L C, θ B,andθ C. r Co/No = Copyright c by Paul Mitiguy 320 Homework 2
5 2.11 Calculating vector cross products with bases. The figure to the right shows a right-handed set of orthogonal unit vectors,, n z.thevectorsu, v, w are defined as: u = n z v = x + y + z n z n w = n z z (a) Use the distributive law for cross products to write u v in terms of,, etc. u v = 2x + 2y + 2z n z (b) Use the definition of the cross product to calculate,, etc. = 0 = n z n z = - = = n z = n z = n z = n z n z = (c) In view of your previous two results, calculate u v. u v = (d) Using the determinant method for calculating the cross product proved in Homework 2.12, calculate u v, u w, and v w. u v = (3z 4 y) + (4x 2 z) + (2y 3 x) n z u w = v w = (e) MG : Modify and submit the following MotionGenesis file to calculate u w and v w. Note: The following commands show the calculation of u v. % File: CalculateCrossProductsWithBasis.al % RigidFrame N Variable x, y, z u> = 2*Nx> + 3*Ny> + 4*Nz> v> = x*nx> + y*ny> + z*nz> w> = 5*Nx> - 6*Ny> + 7*Nz> ucrossv> = Cross( u>, v> ) Save CalculateCrossProductsWithBasis.all Quit n z 2.12 Cross products and determinants. Given right-handed orthogonal unit vectors,, n z and two arbitrary vectors a and b that are expressed in terms of,, n z as shown to the right, prove that calculating a b with the distributive property of the cross product happens to be equal to the determinant of the matrix shown to the right. a = a x + a y + a z n z b = b x + b y + b z n z a b = det n z a x a y a z b x b y b z Copyright c by Paul Mitiguy 321 Homework 2
6 2.13 Optional : Cross product as skew symmetric matrix multiplication. Referring to the previous problem, show that the,, n z coefficients 0 -a z a y of a b happen to be equal to the elements that result from a z 0 -a x the following skew symmetric matrix multiplication. -a y a x 0 After counting the number of computer operations required to multiple the 3 3 matrix by the 3 1 matrix (including multiplication by 0), and comparing the number of operations required to calculate the elements of the simplified answer, it is clear that using a matrix multiplication to calculate a cross product is inefficient True/False Cross products and area calculations. One reason that triangles are important is that complex planar objects can be decomposed into triangles. For example, the polygon B in the figure below can be decomposed into triangles. Knowing the area of two-dimensional objects is helpful in various professions. For example, area measurements are necessary in calculating the acreage and costs associated with building and farming. Knowing the mass properties of a polygon is helpful in determining the motion of two-dimensional objects. B 7 B 5 B 4 r B 1/B 0 = 2.0 b x b y B 8 B 6 r B 2/B 0 = 0.5 b x b y r B 3/B 0 = 3.0 b B x b y 3 r B 4/B 0 = 0.2 b x b y B 2 r B 5/B 0 = -0.5 b B B x b y c B 1 r B 6/B 0 = -1.0 b x b y b y r B 7/B 0 = -2.0 b x b y B 9 bz b r B 8/B 0 = -4.0 b x b y x B 0 r B 9/B 0 = -2.0 b x b y One way to calculate the area of an arbitrary polygon B such as the one shown above is to: Label a vertex B 0 and number the remaining vertices sequentially in a counter-clockwise fashion. Form r B i/b 0, the position vector of vertex B i (i =1, 2,...) from vertex B 0 Calculate A 1, the vector-area of the triangle defined by vertices B 0, B 1,andB 2. Similarly, calculate A 2, A 3,...A 8, the vector-areas of the triangles defined by vertices B 0 B 2 B 3, B 0 B 3 B 4,...B 0 B 8 B 9, respectively. The formula for the vector-area of a triangle is b x b y b z A 1 = 1/2 r B 1/B 0 r B 2/B 0 = 2b z A 2 = 1/2 r B 2/B 0 r B 3/B 0 = A 3 =... = 8.6 b z A 4 =... = A 5 =... = 2.25 b z A 6 =... = 1.5 b z A 7 =... = 9b z A 8 = 1/2 r B 8/B 0 r B 9/B 0 = Calculate A = 8 i=1 A i = The polygon s area is the magnitude of A, i.e., Area = Fill in the previous blanks and determine the polygon s area. Compute cross products with the distributive property (a+b) (c+d) =a c + a d + b c + b d and its definition with the right-hand rule (do not use determinants or look up special formulas in a book). Also, use the fact that b x, b y, b z are orthogonal unit vectors. Copyright c by Paul Mitiguy 322 Homework 2
7 2.15 Scalar triple product with bases. The figure to the right shows a right-handed set of orthogonal unit vectors,, n z.thevectorsu, v, w are defined as: u = n z v = x + y + z n z w = n z n z Calculate u v u, u v w, and u v w. Note: Although the order of operations in u v u is unambiguous, parentheses may clarify your work. u v u = u v w = u v w = 27z 45 x 6 y In view of your last two results, u v w is equal/not equal (circle one) to u v w. It is/is not OK to switch the and in the scalar triple product Optional : Scalar triple products and determinants. Given right-handed orthogonal unit vectors,, n z and three arbitrary vectors a, b, andc that are expressed in terms of,, n z as shown to the right, prove that calculating a (b c) happens to be equal to the determinant of the matrix shown to the right. a = a x + a y + a z n z b = b x + b y + b z n z c = c x + c y + c z n z a x a y a z a b c = b x b y b z c x c y c z 2.17 Constructing unit vectors. Form the unit vector u having the same direction as each vector in the table below. Note: Ensure your answer to the last question agrees with your first two answers, i.e., if c =3 or c = - 3. Vector Unit vector 3-3 Note:,, n z are orthogonal unit vectors n z c n z Copyright c by Paul Mitiguy 323 Homework 2
8 2.18 Locating a microphone (2D). Also see Homework A microphone Q is attached to two pegs B and C by two cables. The point of this practical problem is to determine the distance between Q and point N o knowing the peg locations, cable lengths, and the fact that B, C, Q, andn o all lie in the same plane. Introduce whatever identifiers facilitate your work and try to do the problem first using Euclidean geometry - and then try vectors. Note: There are two mathematical answers to this problem, but one is above the ceiling and requires the cables to be in compression. 8 B 9 15 Q 8 C Quantity Distance from B to C Distance from N o to B Length of cable joining B and Q Length of cable joining C and Q Distance between N o and Q Value 15 m 8m 9m 8m 9.01 m N o 2.19 Locating a microphone (3D). A microphone Q is attached to three pegs A, B, andc by three cables. The point of this practical problem is to determine the distance between Q and point N o knowing the peg locations, cable lengths, and the fact that the walls are perpendicular ( easy problem with the right method). 1 A B N o 13 Q C 8 Quantity Distance from A to B Distance from B to C Distance from N o to B Length of cable joining A and Q Length of cable joining B and Q Length of cable joining C and Q Distance between N o and Q Value 20 m 15 m 8m 15 m 13 m 11 m 13.3 m 2.20 A vector revolution in geometry. The relatively new invention of vectors (Gibbs 1900 AD) has revolutionized Euclidean geometry (Euclid 300 BC). For each geometrical quantity below, circle the vector operation(s) (either the dot-product, cross-product, or both) that is most useful for their calculation. Length: Angle: Area: Volume: 1 Hint: See Section and introduce whatever identifiers facilitate your work. Note: Section shows how to solve nonlinear algebraic equations. This problem can also be solved by-hand. Copyright c by Paul Mitiguy 324 Homework 2
Homework 2. Chapters 1, 2, 3. Vector addition, dot products and cross products
Homework 2. Chapters 1, 2, 3. Vector addition, dot products and cross products 2.1 Dynamics project description Submit a description of your project and include the following: One or more schematics showing
More informationa x b x b y b z Homework 3. Chapter 4. Vector bases and rotation matrices
Homework 3. Chapter 4. Vector bases and rotation matrices 3.1 Dynamics project rotation matrices Update your typed problem description and identifiers as necessary. Include schematics showing the bodies,
More informationChapter 13: Trigonometry Unit 1
Chapter 13: Trigonometry Unit 1 Lesson 1: Radian Measure Lesson 2: Coterminal Angles Lesson 3: Reference Angles Lesson 4: The Unit Circle Lesson 5: Trig Exact Values Lesson 6: Trig Exact Values, Radian
More informationRemark 3.2. The cross product only makes sense in R 3.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationSection 13.4 The Cross Product
Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3-space (but not vectors in 2-space). 1. Basic Definitions
More informationNOTES ON LINEAR ALGEBRA CLASS HANDOUT
NOTES ON LINEAR ALGEBRA CLASS HANDOUT ANTHONY S. MAIDA CONTENTS 1. Introduction 2 2. Basis Vectors 2 3. Linear Transformations 2 3.1. Example: Rotation Transformation 3 4. Matrix Multiplication and Function
More informationRigid Geometric Transformations
Rigid Geometric Transformations Carlo Tomasi This note is a quick refresher of the geometry of rigid transformations in three-dimensional space, expressed in Cartesian coordinates. 1 Cartesian Coordinates
More informationMIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 FALL 2018 KUNIYUKI 150 POINTS TOTAL: 47 FOR PART 1, AND 103 FOR PART
Math 141 Name: MIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 FALL 018 KUNIYUKI 150 POINTS TOTAL: 47 FOR PART 1, AND 103 FOR PART Show all work, simplify as appropriate, and
More informationThe Cross Product. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan The Cross Product
The Cross Product MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Introduction Recall: the dot product of two vectors is a scalar. There is another binary operation on vectors
More informationMathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: Exercise Answers
Mathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: 978-1-118-71219-1 Updated /17/15 Exercise Answers Chapter 1 1. Four right-handed systems: ( i, j, k), ( i, j,
More informationProblem 1: (3 points) Recall that the dot product of two vectors in R 3 is
Linear Algebra, Spring 206 Homework 3 Name: Problem : (3 points) Recall that the dot product of two vectors in R 3 is a x b y = ax + by + cz, c z and this is essentially the same as the matrix multiplication
More informationWorksheet 1.4: Geometry of the Dot and Cross Products
Boise State Math 275 (Ultman) Worksheet 1.4: Geometry of the Dot and Cross Products From the Toolbox (what you need from previous classes): Basic algebra and trigonometry: be able to solve quadratic equations,
More informationVectors. A vector is usually denoted in bold, like vector a, or sometimes it is denoted a, or many other deviations exist in various text books.
Vectors A Vector has Two properties Magnitude and Direction. That s a weirder concept than you think. A Vector does not necessarily start at a given point, but can float about, but still be the SAME vector.
More informationMIPSI: Classic particle pendulum
Chapter 24 MIPSI: Classic particle pendulum The pendulum to the right consists of a small heavy metallic ball tied to a light cable which is connected to the ceiling. This chapter investigates this system
More informationMIPSI: Classic particle pendulum
Chapter 27 MIPSI: Classic particle pendulum (www.motiongenesis.com Textbooks Resources) The pendulum to the right consists of a small heavy metallic ball tied to a light cable which is connected to the
More informationExercise Solutions for Introduction to 3D Game Programming with DirectX 10
Exercise Solutions for Introduction to 3D Game Programming with DirectX 10 Frank Luna, September 6, 009 Solutions to Part I Chapter 1 1. Let u = 1, and v = 3, 4. Perform the following computations and
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.1 Trigonometric Identities Copyright Cengage Learning. All rights reserved. Objectives Simplifying Trigonometric Expressions Proving
More information4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS
4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS MR. FORTIER 1. Trig Functions of Any Angle We now extend the definitions of the six basic trig functions beyond triangles so that we do not have to restrict
More informationLecture 2: Vector-Vector Operations
Lecture 2: Vector-Vector Operations Vector-Vector Operations Addition of two vectors Geometric representation of addition and subtraction of vectors Vectors and points Dot product of two vectors Geometric
More informationMockTime.com. (b) (c) (d)
373 NDA Mathematics Practice Set 1. If A, B and C are any three arbitrary events then which one of the following expressions shows that both A and B occur but not C? 2. Which one of the following is an
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.4 Basic Trigonometric Equations Copyright Cengage Learning. All rights reserved. Objectives Basic Trigonometric Equations Solving
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationPART 1: USING SCIENTIFIC CALCULATORS (50 PTS.)
Math 141 Name: MIDTERM 4 PART 1 (CHAPTERS 5 AND 6: ANALYTIC & MISC. TRIGONOMETRY) MATH 141 SPRING 2018 KUNIYUKI 150 POINTS TOTAL: 50 FOR PART 1, AND 100 FOR PART 2 Show all work, simplify as appropriate,
More informationWhat you will learn today
What you will learn today The Dot Product Equations of Vectors and the Geometry of Space 1/29 Direction angles and Direction cosines Projections Definitions: 1. a : a 1, a 2, a 3, b : b 1, b 2, b 3, a
More informationChapter 2: Statics of Particles
CE297-A09-Ch2 Page 1 Wednesday, August 26, 2009 4:18 AM Chapter 2: Statics of Particles 2.1-2.3 orces as Vectors & Resultants orces are drawn as directed arrows. The length of the arrow represents the
More informationRigid Geometric Transformations
Rigid Geometric Transformations Carlo Tomasi This note is a quick refresher of the geometry of rigid transformations in three-dimensional space, expressed in Cartesian coordinates. 1 Cartesian Coordinates
More informationEUCLIDEAN SPACES AND VECTORS
EUCLIDEAN SPACES AND VECTORS PAUL L. BAILEY 1. Introduction Our ultimate goal is to apply the techniques of calculus to higher dimensions. We begin by discussing what mathematical concepts describe these
More information(arrows denote positive direction)
12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate
More information7.1 Projections and Components
7. Projections and Components As we have seen, the dot product of two vectors tells us the cosine of the angle between them. So far, we have only used this to find the angle between two vectors, but cosines
More information1. Vectors and Matrices
E. 8.02 Exercises. Vectors and Matrices A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by dir A = A, (A 0); A it is the unit vector lying along A and pointed like
More informationQuantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors.
Vectors summary Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors. AB is the position vector of B relative to A and is the vector
More informationBrief Review of Vector Algebra
APPENDIX Brief Review of Vector Algebra A.0 Introduction Vector algebra is used extensively in computational mechanics. The student must thus understand the concepts associated with this subject. The current
More informationAn angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.
Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise
More informationIf the pull is downward (Fig. 1), we want C to point into the page. If the pull is upward (Fig. 2), we want C to point out of the page.
11.5 Cross Product Contemporary Calculus 1 11.5 CROSS PRODUCT This section is the final one about the arithmetic of vectors, and it introduces a second type of vector vector multiplication called the cross
More information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationNotes: Vectors and Scalars
A particle moving along a straight line can move in only two directions and we can specify which directions with a plus or negative sign. For a particle moving in three dimensions; however, a plus sign
More informationVECTORS. Given two vectors! and! we can express the law of vector addition geometrically. + = Fig. 1 Geometrical definition of vector addition
VECTORS Vectors in 2- D and 3- D in Euclidean space or flatland are easy compared to vectors in non- Euclidean space. In Cartesian coordinates we write a component of a vector as where the index i stands
More informationTrigonometric Ratios. θ + k 360
Trigonometric Ratios These notes are intended as a summary of section 6.1 (p. 466 474) in your workbook. You should also read the section for more complete explanations and additional examples. Coterminal
More informationOHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1
OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1 (37) If a bug walks on the sphere x 2 + y 2 + z 2 + 2x 2y 4z 3 = 0 how close and how far can it get from the origin? Solution: Complete
More informationPre-Calculus EOC Review 2016
Pre-Calculus EOC Review 2016 Name The Exam 50 questions, multiple choice, paper and pencil. I. Limits 8 questions a. (1) decide if a function is continuous at a point b. (1) understand continuity in terms
More informationGEOMETRY AND VECTORS
GEOMETRY AND VECTORS Distinguishing Between Points in Space One Approach Names: ( Fred, Steve, Alice...) Problem: distance & direction must be defined point-by-point More elegant take advantage of geometry
More informationMATH 32 FALL 2012 FINAL EXAM - PRACTICE EXAM SOLUTIONS
MATH 3 FALL 0 FINAL EXAM - PRACTICE EXAM SOLUTIONS () You cut a slice from a circular pizza (centered at the origin) with radius 6 along radii at angles 4 and 3 with the positive horizontal axis. (a) (3
More information4 The Trigonometric Functions
Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater
More informationMath Section 4.3 Unit Circle Trigonometry
Math 10 - Section 4. Unit Circle Trigonometry An angle is in standard position if its vertex is at the origin and its initial side is along the positive x axis. Positive angles are measured counterclockwise
More informationExam 1 Review SOLUTIONS
1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More informationWest Windsor-Plainsboro Regional School District Pre-Calculus Grades 11-12
West Windsor-Plainsboro Regional School District Pre-Calculus Grades 11-12 Unit 1: Solving Equations and Inequalities Algebraically and Graphically Content Area: Mathematics Course & Grade Level: Pre Calculus,
More informationSums of Squares (FNS 195-S) Fall 2014
Sums of Squares (FNS 195-S) Fall 014 Record of What We Did Drew Armstrong Vectors When we tried to apply Cartesian coordinates in 3 dimensions we ran into some difficulty tryiing to describe lines and
More informationVectors Year 12 Term 1
Vectors Year 12 Term 1 1 Vectors - A Vector has Two properties Magnitude and Direction - A vector is usually denoted in bold, like vector a, or a, or many others. In 2D - a = xı + yȷ - a = x, y - where,
More informationMATHEMATICS E-23a, Fall 2016 Linear Algebra and Real Analysis I Module #1, Week 2 (Dot and Cross Products, Euclidean Geometry of R n )
MATHEMATICS E-23a, Fall 2016 Linear Algebra and Real Analysis I Module #1, Week 2 (Dot and Cross Products, Euclidean Geometry of R n ) Authors: Paul Bamberg and Kate Penner R scripts by Paul Bamberg Last
More informationMath 5 Trigonometry Final Exam Spring 2009
Math 5 Trigonometry Final Exam Spring 009 NAME Show your work for credit. Write all responses on separate paper. There are 13 problems, all weighted equally. Your 3 lowest scoring answers problem will
More informationGive a geometric description of the set of points in space whose coordinates satisfy the given pair of equations.
1. Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. x + y = 5, z = 4 Choose the correct description. A. The circle with center (0,0, 4)
More informationAS Mathematics Assignment 9 Due Date: Friday 22 nd March 2013
AS Mathematics Assignment 9 Due Date: Friday 22 nd March 2013 NAME GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and submit
More informationEXAM 1. OPEN BOOK AND CLOSED NOTES Thursday, February 18th, 2010
ME 35 - Machine Design I Spring Semester 010 Name of Student Lab. Div. Number EXAM 1. OPEN BOOK AND CLOSED NOTES Thursday, February 18th, 010 Please use the blank paper provided for your solutions. Write
More informationMATH 109 TOPIC 3 RIGHT TRIANGLE TRIGONOMETRY. 3a. Right Triangle Definitions of the Trigonometric Functions
Math 09 Ta-Right Triangle Trigonometry Review Page MTH 09 TOPIC RIGHT TRINGLE TRIGONOMETRY a. Right Triangle Definitions of the Trigonometric Functions a. Practice Problems b. 5 5 90 and 0 60 90 Triangles
More informationDirections: Examine the Unit Circle on the Cartesian Plane (Unit Circle: Circle centered at the origin whose radius is of length 1)
Name: Period: Discovering the Unit Circle Activity Secondary III For this activity, you will be investigating the Unit Circle. You will examine the degree and radian measures of angles. Note: 180 radians.
More informationA2T Trig Packet Unit 1
A2T Trig Packet Unit 1 Name: Teacher: Pd: Table of Contents Day 1: Right Triangle Trigonometry SWBAT: Solve for missing sides and angles of right triangles Pages 1-7 HW: Pages 8 and 9 in Packet Day 2:
More informationMODEL ANSWERS TO HWK #1
MODEL ANSWERS TO HWK # Part B (a The four vertices are (,,, (,,, (,, and (,, The distance between the first two vertices is, since two coordinates differ by There are six edges, corresponding to the choice
More informationFor a semi-circle with radius r, its circumfrence is πr, so the radian measure of a semi-circle (a straight line) is
Radian Measure Given any circle with radius r, if θ is a central angle of the circle and s is the length of the arc sustained by θ, we define the radian measure of θ by: θ = s r For a semi-circle with
More information13 Spherical geometry
13 Spherical geometry Let ABC be a triangle in the Euclidean plane. From now on, we indicate the interior angles A = CAB, B = ABC, C = BCA at the vertices merely by A, B, C. The sides of length a = BC
More informationVectors and 2D Kinematics. AIT AP Physics C
Vectors and 2D Kinematics Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels
More informationConcept Category 3 Trigonometric Functions
Concept Category 3 Trigonometric Functions LT 3A I can prove the addition and subtraction formulas for sine, cosine, and tangent. I can use the addition and subtraction formulas for sine, cosine, and tangent
More informationPURE MATHEMATICS AM 27
AM SYLLABUS (2020) PURE MATHEMATICS AM 27 SYLLABUS 1 Pure Mathematics AM 27 (Available in September ) Syllabus Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics
More informationFE Sta'cs Review. Torch Ellio0 (801) MCE room 2016 (through 2000B door)
FE Sta'cs Review h0p://www.coe.utah.edu/current- undergrad/fee.php Scroll down to: Sta'cs Review - Slides Torch Ellio0 ellio0@eng.utah.edu (801) 587-9016 MCE room 2016 (through 2000B door) Posi'on and
More informationMAT 1339-S14 Class 8
MAT 1339-S14 Class 8 July 28, 2014 Contents 7.2 Review Dot Product........................... 2 7.3 Applications of the Dot Product..................... 4 7.4 Vectors in Three-Space.........................
More informationWarm Up = = 9 5 3) = = ) ) 99 = ) Simplify. = = 4 6 = 2 6 3
Warm Up Simplify. 1) 99 = 3 11 2) 125 + 2 20 = 5 5 + 4 5 = 9 5 3) 2 + 7 2 + 3 7 = 4 + 6 7 + 2 7 + 21 4) 4 42 3 28 = 4 3 3 2 = 4 6 6 = 25 + 8 7 = 2 6 3 Test Results Average Median 5 th : 76.5 78 7 th :
More informationUnit 1 Representing and Operations with Vectors. Over the years you have come to accept various mathematical concepts or properties:
Lesson1.notebook November 27, 2012 Algebra Unit 1 Representing and Operations with Vectors Over the years you have come to accept various mathematical concepts or properties: Communative Property Associative
More informationFeedback D. Incorrect! Exponential functions are continuous everywhere. Look for features like square roots or denominators that could be made 0.
Calculus Problem Solving Drill 07: Trigonometric Limits and Continuity No. of 0 Instruction: () Read the problem statement and answer choices carefully. () Do your work on a separate sheet of paper. (3)
More informationLesson 1: Trigonometry Angles and Quadrants
Trigonometry Lesson 1: Trigonometry Angles and Quadrants An angle of rotation can be determined by rotating a ray about its endpoint or. The starting position of the ray is the side of the angle. The position
More informationNotes on Radian Measure
MAT 170 Pre-Calculus Notes on Radian Measure Radian Angles Terri L. Miller Spring 009 revised April 17, 009 1. Radian Measure Recall that a unit circle is the circle centered at the origin with a radius
More informationREVIEW - Vectors. Vectors. Vector Algebra. Multiplication by a scalar
J. Peraire Dynamics 16.07 Fall 2004 Version 1.1 REVIEW - Vectors By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making
More informationName Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean. A. Definitions: 1.
Name Date Period Notes Formal Geometry Chapter 8 Right Triangles and Trigonometry 8.1 Geometric Mean A. Definitions: 1. Geometric Mean: 2. Right Triangle Altitude Similarity Theorem: If the altitude is
More informationChapter 1. Introduction to Vectors. Po-Ning Chen, Professor. Department of Electrical and Computer Engineering. National Chiao Tung University
Chapter 1 Introduction to Vectors Po-Ning Chen, Professor Department of Electrical and Computer Engineering National Chiao Tung University Hsin Chu, Taiwan 30010, R.O.C. Notes for this course 1-1 A few
More informationWelcome Accelerated Algebra 2! Updates: U8Q1 will be 4/24 Unit Circle Quiz will be 4/24 U8T will be 5/1
Welcome Accelerated Algebra 2! Tear-Out: Pg. 487-492 (Class notes) Updates: U8Q1 will be 4/24 Unit Circle Quiz will be 4/24 U8T will be 5/1 Agenda (1) Warm-Up (2) Review U8H1 (3) Finish Trig Review (4)
More informationF F. proj cos( ) v. v proj v
Geometric Definition of Dot Product 1.2 The Dot Product Suppose you are pulling up on a rope attached to a box, as shown above. How would you find the force moving the box towards you? As stated above,
More informationExample: Inverted pendulum on cart
Chapter 11 Eample: Inverted pendulum on cart The figure to the right shows a rigid body attached by an frictionless pin (revolute) joint to a cart (modeled as a particle). Thecart slides on a horizontal
More informationDepartment of Physics, Korea University
Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an
More informationvand v 3. Find the area of a parallelogram that has the given vectors as adjacent sides.
Name: Date: 1. Given the vectors u and v, find u vand v v. u= 8,6,2, v = 6, 3, 4 u v v v 2. Given the vectors u nd v, find the cross product and determine whether it is orthogonal to both u and v. u= 1,8,
More informationMAC Module 5 Vectors in 2-Space and 3-Space II
MAC 2103 Module 5 Vectors in 2-Space and 3-Space II 1 Learning Objectives Upon completing this module, you should be able to: 1. Determine the cross product of a vector in R 3. 2. Determine a scalar triple
More information6. Vectors. Given two points, P 0 = (x 0, y 0 ) and P 1 = (x 1, y 1 ), a vector can be drawn with its foot at P 0 and
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationIntegrated Math II. IM2.1.2 Interpret given situations as functions in graphs, formulas, and words.
Standard 1: Algebra and Functions Students graph linear inequalities in two variables and quadratics. They model data with linear equations. IM2.1.1 Graph a linear inequality in two variables. IM2.1.2
More informationCHAPTER 4. APPLICATIONS AND REVIEW IN TRIGONOMETRY
CHAPTER 4. APPLICATIONS AND REVIEW IN TRIGONOMETRY In the present chapter we apply the vector algebra and the basic properties of the dot product described in the last chapter to planar geometry and trigonometry.
More informationThe Cross Product. In this section, we will learn about: Cross products of vectors and their applications.
The Cross Product In this section, we will learn about: Cross products of vectors and their applications. THE CROSS PRODUCT The cross product a x b of two vectors a and b, unlike the dot product, is a
More informationHow to rotate a vector using a rotation matrix
How to rotate a vector using a rotation matrix One of the most useful operations in computer graphics is the rotation of a vector using a rotation matrix. I want to introduce the underlying idea of the
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationAlgebra2/Trig Chapter 13 Packet
Algebra2/Trig Chapter 13 Packet In this unit, students will be able to: Use the reciprocal trig identities to express any trig function in terms of sine, cosine, or both. Prove trigonometric identities
More information1.1 Bound and Free Vectors. 1.2 Vector Operations
1 Vectors Vectors are used when both the magnitude and the direction of some physical quantity are required. Examples of such quantities are velocity, acceleration, force, electric and magnetic fields.
More informationUnit Circle. Return to. Contents
Unit Circle Return to Table of Contents 32 The Unit Circle The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant
More informationPrecalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( )
Syllabus Objectives: 5.1 The student will explore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.
More informationIntroduction. Chapter Points, Vectors and Coordinate Systems
Chapter 1 Introduction Computer aided geometric design (CAGD) concerns itself with the mathematical description of shape for use in computer graphics, manufacturing, or analysis. It draws upon the fields
More informationUse estimation strategies reasonably and fluently while integrating content from each of the other strands. PO 1. Recognize the limitations of
for Strand 1: Number and Operations Concept 1: Number Sense Understand and apply numbers, ways of representing numbers, and the relationships among numbers and different number systems. PO 1. Solve problems
More informationCoordinate Systems. Chapter 3. Cartesian Coordinate System. Polar Coordinate System
Chapter 3 Vectors Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels instructions
More information2 Trigonometric functions
Theodore Voronov. Mathematics 1G1. Autumn 014 Trigonometric functions Trigonometry provides methods to relate angles and lengths but the functions we define have many other applications in mathematics..1
More informationMathematics 5 SN TRIGONOMETRY PROBLEMS 2., which one of the following statements is TRUE?, which one of the following statements is TRUE?
Mathematics 5 SN TRIGONOMETRY PROBLEMS 1 If x 4 which one of the following statements is TRUE? A) sin x > 0 and cos x > 0 C) sin x < 0 and cos x > 0 B) sin x > 0 and cos x < 0 D) sin x < 0 and cos x
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationVectors and Plane Geometry
Vectors and Plane Geometry Karl Heinz Dovermann Professor of Mathematics University of Hawaii January 7, 0 Preface During the first week of the semester it is difficult to get started with the course
More informationHonors Algebra 2 Chapter 14 Page 1
Section. (Introduction) Graphs of Trig Functions Objectives:. To graph basic trig functions using t-bar method. A. Sine and Cosecant. y = sinθ y y y y 0 --- --- 80 --- --- 30 0 0 300 5 35 5 35 60 50 0
More informationATHS FC Math Department Al Ain Revision worksheet
ATHS FC Math Department Al Ain Revision worksheet Section Name ID Date Lesson Marks 3.3, 13.1 to 13.6, 5.1, 5.2, 5.3 Lesson 3.3 (Solving Systems of Inequalities by Graphing) Question: 1 Solve each system
More informationFigure 1. A planar mechanism. 1
ME 352 - Machine Design I Summer Semester 201 Name of Student Lab Section Number EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, July 2nd, 201 Use the blank paper provided for your solutions. Write on one
More informationChapter 3. Radian Measure and Circular Functions. Copyright 2005 Pearson Education, Inc.
Chapter 3 Radian Measure and Circular Functions Copyright 2005 Pearson Education, Inc. 3.1 Radian Measure Copyright 2005 Pearson Education, Inc. Measuring Angles Thus far we have measured angles in degrees
More information