Vector Operations. Lecture 19. Robb T. Koether. Hampden-Sydney College. Wed, Oct 7, 2015
|
|
- Marion Alexis Holt
- 5 years ago
- Views:
Transcription
1 Vector Operations Lecture 19 Robb T. Koether Hampden-Sydney College Wed, Oct 7, 2015 Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
2 Outline 1 Magnitude 2 Dot Product 3 Cross Product 4 The vec3 Class 5 Assignment 11 6 Assignment Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
3 Outline 1 Magnitude 2 Dot Product 3 Cross Product 4 The vec3 Class 5 Assignment 11 6 Assignment Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
4 Vector Magnitude Definition (The Dot Product) The magnitude of a vector is its length. It is given by the distance formula. Let v = (v 1, v 2, v 3 ). The magnitude of v, denoted v, is given by v = v1 2 + v v 3 2. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
5 Normalized Vectors To normalize a vector, we divide it by its length. That is, for any vector v 0, the unit vector n with the same direction as v is n = v v. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
6 Outline 1 Magnitude 2 Dot Product 3 Cross Product 4 The vec3 Class 5 Assignment 11 6 Assignment Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
7 The Dot Product Definition (The Dot Product) The dot product of two vectors u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) is defined to be u v = u 1 v 1 + u 2 v 2 + u 3 v 3. Note that the dot product of two vectors is a scalar. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
8 Algebraic Properties of the Dot Product Let u, v, and w be vectors and let c be a real number and let θ be the angle between u and v. Then u v = v u (cu) v = u (cv) = c(u v) u (v + w) = u v + u w v v = v 2 u v = u v cos θ Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
9 Dot Products and Angles A consequence of the last property is that u v > 0 if and only if 0 θ < 90 (acute angle). u v = 0 if and only if θ = 90 (right angle). u v < 0 if and only if 90 < θ 180 (obtuse angle). This is of enormous importance in computer graphics. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
10 Orthogonal Projections Definition (Orthogonal Projection) The orthogonal projection of a vector u onto a vector v is the vector ( u v ) v. v v For example, the projection of u = (5, 0, 2) onto v = (3, 4, 5) is ( u v ) ( ) v = (3, 4, 5) v v ( ) 25 = (3, 4, 5) = 50 ( 3 2, 4 2, 5 2 ). Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
11 Outline 1 Magnitude 2 Dot Product 3 Cross Product 4 The vec3 Class 5 Assignment 11 6 Assignment Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
12 The Cross Product Definition (Cross Product) The cross product of vectors u = (u 1, u 2, u 3 ) and v = (v 1, v 2, v 3 ) is defined to be the vector u v = (u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ). To find normal vectors, we need the cross product. Note that the cross product of vectors is a vector, not a scalar. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
13 The Cross Product u 1 u 2 u 3 v 1 v 2 v 3 An easy way to remember the cross product. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
14 The Cross Product u 1 u 2 u 3 u 1 u 2 v 1 v 2 v 3 v 1 v 2 Duplicate the first and second columns. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
15 The Cross Product u 1 u 2 u 3 u 1 u 2 v 1 v 2 v 3 v 1 v 2 Find this 2 2 determinant for the first component. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
16 The Cross Product u 1 u 2 u 3 u 1 u 2 v 1 v 2 v 3 v 1 v 2 Find the next 2 2 determinant for the second component. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
17 The Cross Product u 1 u 2 u 3 u 1 u 2 v 1 v 2 v 3 v 1 v 2 Find the last 2 2 determinant for the third component. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
18 Algebraic Properties of the Cross Product Let u, v, and w be vectors and let c be a real number and let θ be the angle between u and v. u v = (v u) (cu) v = u (cv) = c(u v) v v = 0 (u v) u = (u v) v = 0 u v = u v sin θ Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
19 The Right-hand Rule The right-hand rule helps us remember which way u v points. Arrange the thumb, index finger, and middle finger so that they are mutually orthogonal. Let the thumb represent u and the index finger represent v. Then the middle finger represents u v. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
20 Finding Surface Normals Example (Finding Surface Normals) Given a triangle ABC, where A = (1, 1, 2), B = (3, 1, 5), and C = (1, 0, 4), find a unit vector that is normal to the surface. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
21 Example Example (Finding Surface Normals) Let Then w = u v = (3, 4, 2). u = B A = (2, 0, 3) v = C A = (0, 1, 2) w = 29, so the unit normal is ( 3 n =, 4, 2 ) Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
22 Outline 1 Magnitude 2 Dot Product 3 Cross Product 4 The vec3 Class 5 Assignment 11 6 Assignment Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
23 The vec3 Class Vector Functions float length(vecn v); float dot(vecn u, vecn v); vec3 cross(vecn u, vecn v); In the vec classes (vec2,vec3,vec4), there are member functions for the length, the dot product. The cross product applies to vec3 objects only. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
24 Outline 1 Magnitude 2 Dot Product 3 Cross Product 4 The vec3 Class 5 Assignment 11 6 Assignment Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
25 Assignment 11 Assignment 11 See handout. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
26 Outline 1 Magnitude 2 Dot Product 3 Cross Product 4 The vec3 Class 5 Assignment 11 6 Assignment Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
27 Assignment Assignment Read pp. xxx - xxx, xxx. Robb T. Koether (Hampden-Sydney College) Vector Operations Wed, Oct 7, / 23
Direct Proof Floor and Ceiling
Direct Proof Floor and Ceiling Lecture 17 Section 4.5 Robb T. Koether Hampden-Sydney College Wed, Feb 12, 2014 Robb T. Koether (Hampden-Sydney College) Direct Proof Floor and Ceiling Wed, Feb 12, 2014
More informationComposition of Functions
Composition of Functions Lecture 34 Section 7.3 Robb T. Koether Hampden-Sydney College Mon, Mar 25, 2013 Robb T. Koether (Hampden-Sydney College) Composition of Functions Mon, Mar 25, 2013 1 / 29 1 Composition
More informationInstantaneous Rate of Change
Instantaneous Rate of Change Lecture 13 Section 2.1 Robb T. Koether Hampden-Sydney College Wed, Feb 8, 2017 Robb T. Koether (Hampden-Sydney College) Instantaneous Rate of Change Wed, Feb 8, 2017 1 / 11
More informationMaximums and Minimums
Maximums and Minimums Lecture 25 Section 3.1 Robb T. Koether Hampden-Sydney College Mon, Mar 6, 2017 Robb T. Koether (Hampden-Sydney College) Maximums and Minimums Mon, Mar 6, 2017 1 / 9 Objectives Objectives
More informationThe Interpretation of λ
The Interpretation of λ Lecture 49 Section 7.5 Robb T. Koether Hampden-Sydney College Wed, Apr 26, 2017 Robb T. Koether (Hampden-Sydney College) The Interpretation of λ Wed, Apr 26, 2017 1 / 6 Objectives
More informationDirect Proof Division into Cases
Direct Proof Division into Cases Lecture 16 Section 4.4 Robb T. Koether Hampden-Sydney College Mon, Feb 10, 2014 Robb T. Koether (Hampden-Sydney College) Direct Proof Division into Cases Mon, Feb 10, 2014
More informationThe Pairwise-Comparison Method
The Pairwise-Comparison Method Lecture 12 Section 1.5 Robb T. Koether Hampden-Sydney College Mon, Sep 19, 2016 Robb T. Koether (Hampden-Sydney College) The Pairwise-Comparison Method Mon, Sep 19, 2016
More informationDirect Proof Universal Statements
Direct Proof Universal Statements Lecture 13 Section 4.1 Robb T. Koether Hampden-Sydney College Wed, Feb 6, 2013 Robb T. Koether (Hampden-Sydney College) Direct Proof Universal Statements Wed, Feb 6, 2013
More informationDirect Proof Rational Numbers
Direct Proof Rational Numbers Lecture 14 Section 4.2 Robb T. Koether Hampden-Sydney College Thu, Feb 7, 2013 Robb T. Koether (Hampden-Sydney College) Direct Proof Rational Numbers Thu, Feb 7, 2013 1 /
More informationCompound Interest. Lecture 34 Section 4.1. Robb T. Koether. Hampden-Sydney College. Tue, Mar 28, 2017
Compound Interest Lecture 34 Section 4.1 Robb T. Koether Hampden-Sydney College Tue, Mar 28, 2017 Robb T. Koether (Hampden-Sydney College) Compound Interest Tue, Mar 28, 2017 1 / 8 Reminder Reminder Test
More informationPredicates and Quantifiers
Predicates and Quantifiers Lecture 9 Section 3.1 Robb T. Koether Hampden-Sydney College Wed, Jan 29, 2014 Robb T. Koether (Hampden-Sydney College) Predicates and Quantifiers Wed, Jan 29, 2014 1 / 32 1
More informationNondeterministic Finite Automata
Nondeterministic Finite Automata Lecture 6 Section 2.2 Robb T. Koether Hampden-Sydney College Mon, Sep 5, 2016 Robb T. Koether (Hampden-Sydney College) Nondeterministic Finite Automata Mon, Sep 5, 2016
More informationPaired Samples. Lecture 37 Sections 11.1, 11.2, Robb T. Koether. Hampden-Sydney College. Mon, Apr 2, 2012
Paired Samples Lecture 37 Sections 11.1, 11.2, 11.3 Robb T. Koether Hampden-Sydney College Mon, Apr 2, 2012 Robb T. Koether (Hampden-Sydney College) Paired Samples Mon, Apr 2, 2012 1 / 17 Outline 1 Dependent
More informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether Hampden-Sydney College Fri, Feb 8, 2013 Robb T. Koether (Hampden-Sydney College) Direct Proof Divisibility Fri, Feb 8, 2013 1 / 20 1 Divisibility
More informationDirect Proof Divisibility
Direct Proof Divisibility Lecture 15 Section 4.3 Robb T. Koether Hampden-Sydney College Fri, Feb 7, 2014 Robb T. Koether (Hampden-Sydney College) Direct Proof Divisibility Fri, Feb 7, 2014 1 / 23 1 Divisibility
More informationClosure Properties of Regular Languages
Closure Properties of Regular Languages Lecture 13 Section 4.1 Robb T. Koether Hampden-Sydney College Wed, Sep 21, 2016 Robb T. Koether (Hampden-Sydney College) Closure Properties of Regular Languages
More informationConfidence Intervals for Proportions Sections 22.2, 22.3
Confidence Intervals for Proportions Sections 22.2, 22.3 Lecture 41 Robb T. Koether Hampden-Sydney College Mon, Apr 4, 2016 Robb T. Koether (Hampden-Sydney College)Confidence Intervals for ProportionsSections
More informationConditional Statements
Conditional Statements Lecture 3 Section 2.2 Robb T. Koether Hampden-Sydney College Fri, Jan 17, 2014 Robb T. Koether (Hampden-Sydney College) Conditional Statements Fri, Jan 17, 2014 1 / 26 1 Conditional
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 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 informationSampling Distribution of a Sample Proportion
Sampling Distribution of a Sample Proportion Lecture 26 Section 8.4 Robb T. Koether Hampden-Sydney College Mon, Oct 10, 2011 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion
More informationThe Cobb-Douglas Production Functions
The Cobb-Douglas Production Functions Lecture 40 Section 7.1 Robb T. Koether Hampden-Sydney College Mon, Apr 10, 2017 Robb T. Koether (Hampden-Sydney College) The Cobb-Douglas Production Functions Mon,
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 informationNegations of Quantifiers
Negations of Quantifiers Lecture 10 Section 3.2 Robb T. Koether Hampden-Sydney College Thu, Jan 31, 2013 Robb T. Koether (Hampden-Sydney College) Negations of Quantifiers Thu, Jan 31, 2013 1 / 20 1 Negations
More informationStrong Mathematical Induction
Strong Mathematical Induction Lecture 23 Section 5.4 Robb T. Koether Hampden-Sydney College Mon, Feb 24, 2014 Robb T. Koether (Hampden-Sydney College) Strong Mathematical Induction Mon, Feb 24, 2014 1
More informationSampling Distribution of a Sample Proportion
Sampling Distribution of a Sample Proportion Lecture 26 Section 8.4 Robb T. Koether Hampden-Sydney College Mon, Mar 1, 2010 Robb T. Koether (Hampden-Sydney College) Sampling Distribution of a Sample Proportion
More informationElementary Logic and Proof
Elementary Logic and Proof Lecture 5 Robb T. Koether Hampden-Sydney College Mon, Feb 6, 2017 Robb T. Koether (Hampden-Sydney College) Elementary Logic and Proof Mon, Feb 6, 2017 1 / 33 Outline 1 Statements
More informationIndependent Samples: Comparing Means
Independent Samples: Comparing Means Lecture 38 Section 11.4 Robb T. Koether Hampden-Sydney College Fri, Nov 4, 2011 Robb T. Koether (Hampden-Sydney College) Independent Samples:Comparing Means Fri, Nov
More informationBinary Search Trees. Lecture 29 Section Robb T. Koether. Hampden-Sydney College. Fri, Apr 8, 2016
Binary Search Trees Lecture 29 Section 19.2 Robb T. Koether Hampden-Sydney College Fri, Apr 8, 2016 Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, 2016 1 / 40 1 Binary Search
More informationSection 8.2 Vector Angles
Section 8.2 Vector Angles INTRODUCTION Recall that a vector has these two properties: 1. It has a certain length, called magnitude 2. It has a direction, indicated by an arrow at one end. In this section
More informationSection 10.7 The Cross Product
44 Section 10.7 The Cross Product Objective #0: Evaluating Determinants. Recall the following definition for determinants: Determinants a The determinant for matrix 1 b 1 is denoted as a 1 b 1 a b a b
More informationR n : The Cross Product
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler R n : The Cross Product Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)
More informationLinear Algebra. Alvin Lin. August December 2017
Linear Algebra Alvin Lin August 207 - December 207 Linear Algebra The study of linear algebra is about two basic things. We study vector spaces and structure preserving maps between vector spaces. A vector
More information10.2,3,4. Vectors in 3D, Dot products and Cross Products
Name: Section: 10.2,3,4. Vectors in 3D, Dot products and Cross Products 1. Sketch the plane parallel to the xy-plane through (2, 4, 2) 2. For the given vectors u and v, evaluate the following expressions.
More informationObjective 1. Lesson 87: The Cross Product of Vectors IBHL - SANTOWSKI FINDING THE CROSS PRODUCT OF TWO VECTORS
Lesson 87: The Cross Product of Vectors IBHL - SANTOWSKI In this lesson you will learn how to find the cross product of two vectors how to find an orthogonal vector to a plane defined by two vectors how
More informationVectors and Matrices Lecture 2
Vectors and Matrices Lecture 2 Dr Mark Kambites School of Mathematics 13/03/2014 Dr Mark Kambites (School of Mathematics) COMP11120 13/03/2014 1 / 20 How do we recover the magnitude of a vector from its
More informationWorksheet 1.3: Introduction to the Dot and Cross Products
Boise State Math 275 (Ultman Worksheet 1.3: Introduction to the Dot and Cross Products From the Toolbox (what you need from previous classes Trigonometry: Sine and cosine functions. Vectors: Know what
More information12.1 Three Dimensional Coordinate Systems (Review) Equation of a sphere
12.2 Vectors 12.1 Three Dimensional Coordinate Systems (Reiew) Equation of a sphere x a 2 + y b 2 + (z c) 2 = r 2 Center (a,b,c) radius r 12.2 Vectors Quantities like displacement, elocity, and force inole
More informationThe Cross Product -(10.4)
The Cross Product -(10.4) Questions: What is the definition of the cross product of two vectors? Is it a scalar or vector? What can you say about vectors u and v (all possibilities) if the cross product
More informationBrief Review of Exam Topics
Math 32A Discussion Session Week 3 Notes October 17 and 19, 2017 We ll use this week s discussion session to prepare for the first midterm. We ll start with a quick rundown of the relevant topics, and
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 informationChapter 6 Additional Topics in Trigonometry, Part II
Chapter 6 Additional Topics in Trigonometry, Part II Section 3 Section 4 Section 5 Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number Vocabulary Directed line segment
More informationInner Product Spaces 6.1 Length and Dot Product in R n
Inner Product Spaces 6.1 Length and Dot Product in R n Summer 2017 Goals We imitate the concept of length and angle between two vectors in R 2, R 3 to define the same in the n space R n. Main topics are:
More informationReview of Coordinate Systems
Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate
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 informationMatrix Operations. Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix
Linear Combination Vector Algebra Angle Between Vectors Projections and Reflections Equality of matrices, Augmented Matrix Matrix Operations Matrix Addition and Matrix Scalar Multiply Matrix Multiply Matrix
More informationThe Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /
The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring 2012 1 / 15 Introduction The cross product is the second multiplication operation between vectors we will
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 informationLinear Algebra: Homework 3
Linear Algebra: Homework 3 Alvin Lin August 206 - December 206 Section.2 Exercise 48 Find all values of the scalar k for which the two vectors are orthogonal. [ ] [ ] 2 k + u v 3 k u v 0 2(k + ) + 3(k
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 information6.4 Vectors and Dot Products
6.4 Vectors and Dot Products Copyright Cengage Learning. All rights reserved. What You Should Learn Find the dot product of two vectors and use the properties of the dot product. Find the angle between
More informationElectromagnetic Field Theory
University of Babylon College of Engineering Department of Electrical Engineering Class :2 st Lectures of Electromagnetic Field Theory By ssistant lecturer : hmed Hussein Shatti l-isawi 2014-2015 References
More informationVector Operations. Vector Operations. Graphical Operations. Component Operations. ( ) ˆk
Vector Operations Vector Operations ME 202 Multiplication by a scalar Addition/subtraction Scalar multiplication (dot product) Vector multiplication (cross product) 1 2 Graphical Operations Component Operations
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 informationDot product. The dot product is an inner product on a coordinate vector space (Definition 1, Theorem
Dot product The dot product is an inner product on a coordinate vector space (Definition 1, Theorem 1). Definition 1 Given vectors v and u in n-dimensional space, the dot product is defined as, n v u v
More informationMathematics 2203, Test 1 - Solutions
Mathematics 220, Test 1 - Solutions F, 2010 Philippe B. Laval Name 1. Determine if each statement below is True or False. If it is true, explain why (cite theorem, rule, property). If it is false, explain
More informationPractical Linear Algebra: A Geometry Toolbox
Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 4: Changing Shapes: Linear Maps in 2D Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/pla
More informationComputer Graphics MTAT Raimond Tunnel
Computer Graphics MTAT.03.015 Raimond Tunnel Points and Vectors In computer graphics we distinguish: Point a location in space (location vector, kohavektor) Vector a direction in space (direction vector,
More informationMTH 2310, FALL Introduction
MTH 2310, FALL 2011 SECTION 6.2: ORTHOGONAL SETS Homework Problems: 1, 5, 9, 13, 17, 21, 23 1, 27, 29, 35 1. Introduction We have discussed previously the benefits of having a set of vectors that is linearly
More informationMath 32A Discussion Session Week 2 Notes October 10 and 12, 2017
Math 32A Discussion Session Week 2 Notes October 10 and 12, 2017 Since we didn t get a chance to discuss parametrized lines last week, we may spend some time discussing those before moving on to the dot
More informationVECTORS IN THREE DIMENSIONS
1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude
More informationWorksheet for Lecture 23 (due December 4) Section 6.1 Inner product, length, and orthogonality
Worksheet for Lecture (due December 4) Name: Section 6 Inner product, length, and orthogonality u Definition Let u = u n product or dot product to be and v = v v n be vectors in R n We define their inner
More informationLecture 26 Section 8.4. Wed, Oct 14, 2009
PDFs n = Lecture 26 Section 8.4 Hampden-Sydney College Wed, Oct 14, 2009 Outline PDFs n = 1 2 PDFs n = 3 4 5 6 Outline PDFs n = 1 2 PDFs n = 3 4 5 6 PDFs n = Exercise 8.12, page 528. Suppose that 60% of
More information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More informationThe Cross Product The cross product of v = (v 1,v 2,v 3 ) and w = (w 1,w 2,w 3 ) is
The Cross Product 1-1-2018 The cross product of v = (v 1,v 2,v 3 ) and w = (w 1,w 2,w 3 ) is v w = (v 2 w 3 v 3 w 2 )î+(v 3 w 1 v 1 w 3 )ĵ+(v 1 w 2 v 2 w 1 )ˆk = v 1 v 2 v 3 w 1 w 2 w 3. Strictly speaking,
More informationQuiz 2 Practice Problems
Quiz Practice Problems Practice problems are similar, both in difficulty and in scope, to the type of problems you will see on the quiz. Problems marked with a are for your entertainment and are not essential.
More informationOmm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
More informationin Trigonometry Name Section 6.1 Law of Sines Important Vocabulary
Name Chapter 6 Additional Topics in Trigonometry Section 6.1 Law of Sines Objective: In this lesson you learned how to use the Law of Sines to solve oblique triangles and how to find the areas of oblique
More informationMATH 19520/51 Class 2
MATH 19520/51 Class 2 Minh-Tam Trinh University of Chicago 2017-09-27 1 Review dot product. 2 Angles between vectors and orthogonality. 3 Projection of one vector onto another. 4 Cross product and its
More informationLecture 10: Vector Algebra: Orthogonal Basis
Lecture 0: Vector Algebra: Orthogonal Basis Orthogonal Basis of a subspace Computing an orthogonal basis for a subspace using Gram-Schmidt Orthogonalization Process Orthogonal Set Any set of vectors that
More informationSolution to Homework 1
Solution to Homework 1 Olena Bormashenko September, 0 Section 1.1:, 5b)d), 7b)f), 8a)b), ; Section 1.:, 5, 7, 9, 10, b)c), 1, 15b), 3 Section 1.1. In each of the following cases, find a point that is two-thirds
More informationMATH 12 CLASS 4 NOTES, SEP
MATH 12 CLASS 4 NOTES, SEP 28 2011 Contents 1. Lines in R 3 1 2. Intersections of lines in R 3 2 3. The equation of a plane 4 4. Various problems with planes 5 4.1. Intersection of planes with planes or
More informationMATH 12 CLASS 2 NOTES, SEP Contents. 2. Dot product: determining the angle between two vectors 2
MATH 12 CLASS 2 NOTES, SEP 23 2011 Contents 1. Dot product: definition, basic properties 1 2. Dot product: determining the angle between two vectors 2 Quick links to definitions/theorems Dot product definition
More informationMath The Dot Product
Math 213 - The Dot Product Peter A. Perry University of Kentucky August 26, 2018 Homework Webwork A1 is due Wednesday night Re-read section 12.3, pp. 807 812 Begin work on problems 1-37 (odd), 41-51 (odd)
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 informationMath 276, Spring 2007 Additional Notes on Vectors
Math 276, Spring 2007 Additional Notes on Vectors 1.1. Real Vectors. 1. Scalar Products If x = (x 1,..., x n ) is a vector in R n then the length of x is x = x 2 1 + + x2 n. We sometimes use the notation
More informationLecture 11 Sections 4.5, 4.7. Wed, Feb 18, 2009
The s s The s Lecture 11 Sections 4.5, 4.7 Hampden-Sydney College Wed, Feb 18, 2009 Outline The s s 1 s 2 3 4 5 6 The LR(0) Parsing s The s s There are two tables that we will construct. The action table
More informationVectors Coordinate frames 2D implicit curves 2D parametric curves. Graphics 2008/2009, period 1. Lecture 2: vectors, curves, and surfaces
Graphics 2008/2009, period 1 Lecture 2 Vectors, curves, and surfaces Computer graphics example: Pixar (source: http://www.pixar.com) Computer graphics example: Pixar (source: http://www.pixar.com) Computer
More informationCHAPTER 10 VECTORS POINTS TO REMEMBER
For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two
More informationCentral to this is two linear transformations: the Fourier Transform and the Laplace Transform. Both will be considered in later lectures.
In this second lecture, I will be considering signals from the frequency perspective. This is a complementary view of signals, that in the frequency domain, and is fundamental to the subject of signal
More informationDot Product August 2013
Dot Product 12.3 30 August 2013 Dot product. v = v 1, v 2,..., v n, w = w 1, w 2,..., w n The dot product v w is v w = v 1 w 1 + v 2 w 2 + + v n w n n = v i w i. i=1 Example: 1, 4, 5 2, 8, 0 = 1 2 + 4
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 information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
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 informationAppendix. Vectors, Systems of Equations
ppendix Vectors, Systems of Equations Vectors, Systems of Equations.1.1 Vectors Scalar physical quantities (e.g., time, mass, density) possess only magnitude. Vectors are physical quantities (e.g., force,
More informationLecture 5 3D polygonal modeling Part 1: Vector graphics Yong-Jin Liu.
Fundamentals of Computer Graphics Lecture 5 3D polygonal modeling Part 1: Vector graphics Yong-Jin Liu liuyongjin@tsinghua.edu.cn Material by S.M.Lea (UNC) Introduction In computer graphics, we work with
More informationMath 261 Lecture Notes: Sections 6.1, 6.2, 6.3 and 6.4 Orthogonal Sets and Projections
Math 6 Lecture Notes: Sections 6., 6., 6. and 6. Orthogonal Sets and Projections We will not cover general inner product spaces. We will, however, focus on a particular inner product space the inner product
More informationThe Cross Product of Two Vectors
The Cross roduct of Two Vectors In proving some statements involving surface integrals, there will be a need to approximate areas of segments of the surface by areas of parallelograms. Therefore it is
More informationVectors and the Geometry of Space
Vectors and the Geometry of Space Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized by a single real number scaled to appropriate units of
More informationIntro Vectors 2D implicit curves 2D parametric curves. Graphics 2011/2012, 4th quarter. Lecture 2: vectors, curves, and surfaces
Lecture 2, curves, and surfaces Organizational remarks Tutorials: Tutorial 1 will be online later today TA sessions for questions start next week Practicals: Exams: Make sure to find a team partner very
More informationMath 3191 Applied Linear Algebra
Math 191 Applied Linear Algebra Lecture 1: Inner Products, Length, Orthogonality Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/ Motivation Not all linear systems have
More informationLecture 1.4: Inner products and orthogonality
Lecture 1.4: Inner products and orthogonality Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M.
More informationCourse Name : Physics I Course # PHY 107
Course Name : Physics I Course # PHY 107 Lecture-2 : Representation of Vectors and the Product Rules Abu Mohammad Khan Department of Mathematics and Physics North South University http://abukhan.weebly.com
More informationUniversity of Sheffield. PHY120 - Vectors. Dr Emiliano Cancellieri
University of Sheffield PHY120 - Vectors Dr Emiliano Cancellieri October 14, 2015 Contents 1 Lecture 1 2 1.1 Basic concepts of vectors........................ 2 1.2 Cartesian components of vectors....................
More informationCoordinates and vectors. Background mathematics review
Coordinates and vectors Background mathematics review David Miller Coordinates and vectors Coordinate axes and vectors Background mathematics review David Miller y Coordinate axes and vectors z x Ordinary
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationCS 4300 Computer Graphics. Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011
CS 4300 Computer Graphics Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011 October 8, 2011 College of Computer and Information Science, Northeastern Universit 1 Toda s Topics Linear Algebra Review
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 informationThe Dot Product
The Dot Product 1-9-017 If = ( 1,, 3 ) and = ( 1,, 3 ) are ectors, the dot product of and is defined algebraically as = 1 1 + + 3 3. Example. (a) Compute the dot product (,3, 7) ( 3,,0). (b) Compute the
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 information