Scalar multiplication and algebraic direction of a vector
|
|
- Sabrina Spencer
- 6 years ago
- Views:
Transcription
1 Roberto s Notes on Linear Algebra Chapter 1: Geometric ectors Section 5 Scalar multiplication and algebraic direction of a ector What you need to know already: of a geometric ectors. Length and geometric direction of a ector. What you can learn here: The operation of scalar multiplication. How to use this operation to construct an efficient definition of direction for a ector. According to our definitions, a 2D ector is simply an ordered pair of usual numbers and a 3D ector is an ordered triple of numbers. Although in each case we can deise and use a geometric representation, the algebraic aspect is the key feature that will be exploited in the rest of the course. As a way of introduction, I will now show you how, by using the arithmetic operations that are so familiar to us from grade school, we can deelop a useful operation for ectors that will lead to a much better way to identify a direction. The scalar multiplication of a scalar c and a ector is performed by multiplying each of the components of by c. Therefore: In 2D: c c c c In 3D: c c c c c Two ectors u and are scalar multiples of each other if there is a scalar c such that either c or u c. Example: By this definition: and It turns out that this is an extremely useful operation, one that forms the backbone for most of the material of this book. To start showing you how useful this is, here is the better definition of direction that I hae promised: u Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 1
2 Proof Technical fact Two non-zero geometric ectors hae the same direction if and only if they are scalar multiples of each other. Moreoer, they hae the same orientation if such scalars are positie. Same direction Scalar multiples: In 2D, if two ectors and w w 1 2 w hae the same 1 2 direction, it means that either they are both ertical, or they hae the same slope. If they are both ertical, they are of the form 0 2 and w 0 w 2 with both 2 and w 2 different from 0. In that case, if we w2 pick c we see that the two ectors are scalar multiples of each 2 other. Scalar multiples Same direction: If, w w 1 2 w and w c, it means that 1 2 w2 c2 2 w c1 c2 and hence, so that and w hae the w c same direction. This argument only fails if 1 0, but in that case w1 0 as well and hence both ectors are ertical and so they hae the same direction. I leae to you the proof for the 3D case. You want me to proe it in the more difficult situation? That s rich! And why do I need to be able to do proofs anyway? I trust what you say I am honored by your trust, but learning how to do simple proofs is one of the goals of this course, as well as a skill that will be ery useful to you in the future. So we all expect you to learn how to do it. As for the difficulty leel, in 3D the proof is the same, just longer, since there are more proportions to juggle, but the procedure is the same, so you ll get a good workout. In any case, now that we hae a more workable definition of direction, we can look at another word with which you are familiar and that we can now define formally: Two geometric ectors are parallel if they hae the same direction, or, equialently, if they are scalar multiples of each other. Example: If we consider the ectors 6 4 b 96, we a and 2 notice that b 1.5a, or a b. 3 This means that they are multiples of each other and hence they hae the same direction, or are parallel, as the picture clearly shows. a b Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 2
3 It looks like two ectors are parallel if one of them is a stretched ersion of the other. That is correct, since we are maintaining the direction. But there is more! If you look at the picture you may get the impression that b is about 1.5 times longer than a. This may seem obious, but remember that the scalar multiplication of the definition occurs on the components, not the length, so we need to check if this obseration is in fact true and not a coincidence: Proof Technical fact Scalar multiplication has the effect of changing the length of the ector by a factor equal to the absolute alue of the scalar factor: c Here is how to proe this fact in 2D: c c c c c c c c c c c 1 2 c Again I leae you the fun of proing this in 3. Yes, and in fact this gies us a ery useful way to obtain a unit ector in any direction we want. Technical fact Gien any non-zero ector, the ector 1 denoted by e, it is a unit ector and has the same direction and orientation as. The process of constructing a unit ector in the direction of a gien non-zero ector is called normalization. Example: If we normalize the ector 3 2 obtain: e Both ectors are shown in the picture and so is the ector 1 e 3 2, which has the same direction, but opposite 13 orientation we e e is So, we can use scalar multiplication to lengthen and shorten ectors at will, correct? I hae a question about jargon and one about notation. Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 3
4 Excellent! Both are ery important aspects of this course, so it is best to clarify them as soon as they arise. First of all, we called this operation scalar multiplication. Why not the shorter scalar product? It is fine to refer to this operation as the scalar product of a scalar and a ector, but later we shall meet another operation that is also called scalar product, but refers to a rather different situation and produces a different result. So, we can use this terminology as long as the context is clear. Fair enough. Now, when we multiply a ector by a fraction, can we make the notation smaller by placing the ector in the numerator? Absolutely! This is a ery conenient moe and we might as well formalize it in a box. A scalar diision of a ector by a scalar is obtained by multiplying the ector by the reciprocal of the scalar. That is: 1 c c When defining what we mean by two ectors being multiples of each other, I excluded consideration of the zero ector, for otherwise we may hae generated a diision by 0 in the formula. That is also reasonable because the zero ector does not hae a proper direction (it includes a single point) and because the constant that would change any ector to the zero ector is 0, but one cannot diide by zero in order to reerse the relation. Although reasonable, it turns out that this exclusion would gie us lots of headaches in future deelopments. To aoid this problem, we shall use the following conention: The zero ector is considered to be a multiple of any other ector, hence parallel to any other ector, despite the fact that it does not hae a direction and that the algebraic relationship of being scalar multiple would only be satisfied in one direction. I am delaying examples of this definition because at this point their simplicity would offend your intelligence and because we shall hae many opportunities to see it in action when it counts. For now, just keep it in mind and write down any questions you may hae about it. Before we close this chapter and moe to bigger and better things, I want to clarify a little pedantic point that may bother us later if we don t. Summary Since scalar multiplication changes each component by the same factor, it does not affect the direction of a ector. Therefore, it can be used to identify such direction through an equialence relation. Two ectors hae the same direction, or are parallel, if they are multiples of each other. Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 4
5 A unit ector can be seen as being representatie of a direction. Common errors to aoid Distinguish between direction and orientation: they are related concepts, but not the same thing! Remember that the zero ector is considered as being parallel to any other ector, een though this slightly iolates the mutual requirement of the definition. Learning questions for Section LA 1-5 Reiew questions: 1. Describe how scalar multiplication works. 3. Discuss the role of unit ectors in the concept of algebraic direction of a ector. 2. Explain how scalar multiplication is used to define the direction of a geometric ector. Memory questions: 1. Which formula defines the product of a scalar and a 3D ector? 2. Which formula defines the unit ector with the same direction and orientation of a gien ector? Computation questions: 1. Gien the ectors 3 5 and u 5.2 3, determine 3 and -2u. 2. Determine the unit ectors parallel to 3 5 and u Which ector is the unit ector in the direction of the ector PQ joining P 3, 2 1, 2? and Q Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 5
6 4. Determine the unit ector of that has the opposite orientation of the ector Determine the magnitude and the direction as a unit ector of the ector whose tail is at the point 3,1, 0 and whose tip is at 1, 2, Find all alues of k for which the ectors k k 2, 3 k 1 parallel. 7. Gien the ectors u 12kand u are, determine the alues of k for which, respectiely: a) u and are parallel and with the same orientation; b) u and are parallel and with opposite orientation. Theory questions: 1. How is a unit ector obtained from a gien ector? 2. What is the main purpose of unit ectors? 3. What is the geometric effect of multiplying a ector by a scalar? 4. Gien a ector abc, construct a ector w that has half its length. 5. How many unit ectors hae the same direction as a gien ector? 6. Which are the unit ectors in the direction of the line y x? Proof questions: 1. Show that if two 2D ectors a1 a 2 and 1 2 a 1 b1 0, then they hae the same orientation. bb hae the same direction and 2. Proe that scalar multiplication is distributie, that is, for any geometric ectors u and and any scalars h and k, h k u hu ku. h u hu h and 3. Proe that scalar multiplication is associatie, that is, for any geometric ector u h ku hk u. and scalars h and k, 4. Proe that the scalar 1 is an identity for scalar multiplication, that is, for any geometric ector, 1. Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 6
7 Templated questions: In these questions, make your own choice of any items inoled. 1. Choose a 2D ector and then construct the unit ector parallel to it and with the same orientation. 2. Choose a 3D ector and then construct the unit ector parallel to it and with the same orientation. 3. Choose a 2D ector and then construct the unit ector parallel to it and with the opposite orientation. 4. Choose a 3D ector and then construct the unit ector parallel to it and with the opposite orientation. What questions do you hae for your instructor? Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 7
8 Linear Algebra Chapter 1: Geometric ectors Section 5: Scalar multiplication and algebraic direction of a ector Page 8
Roberto s Notes on Linear Algebra Chapter 1: Geometric vectors Section 8. The dot product
Roberto s Notes on Linear Algebra Chapter 1: Geometric ectors Section 8 The dot product What you need to know already: What a linear combination of ectors is. What you can learn here: How to use two ectors
More informationLECTURE 2: CROSS PRODUCTS, MULTILINEARITY, AND AREAS OF PARALLELOGRAMS
LECTURE : CROSS PRODUCTS, MULTILINEARITY, AND AREAS OF PARALLELOGRAMS MA1111: LINEAR ALGEBRA I, MICHAELMAS 016 1. Finishing up dot products Last time we stated the following theorem, for which I owe you
More informationDefinition of geometric vectors
Roberto s Notes on Linear Algebra Chapter 1: Geometric vectors Section 2 of geometric vectors What you need to know already: The general aims behind the concept of a vector. What you can learn here: The
More informationUNDERSTAND MOTION IN ONE AND TWO DIMENSIONS
SUBAREA I. COMPETENCY 1.0 UNDERSTAND MOTION IN ONE AND TWO DIMENSIONS MECHANICS Skill 1.1 Calculating displacement, aerage elocity, instantaneous elocity, and acceleration in a gien frame of reference
More informationA Geometric Review of Linear Algebra
A Geometric Reiew of Linear Algebra The following is a compact reiew of the primary concepts of linear algebra. The order of presentation is unconentional, with emphasis on geometric intuition rather than
More informationdifferent formulas, depending on whether or not the vector is in two dimensions or three dimensions.
ectors The word ector comes from the Latin word ectus which means carried. It is best to think of a ector as the displacement from an initial point P to a terminal point Q. Such a ector is expressed as
More informationVISUAL PHYSICS ONLINE RECTLINEAR MOTION: UNIFORM ACCELERATION
VISUAL PHYSICS ONLINE RECTLINEAR MOTION: UNIFORM ACCELERATION Predict Obsere Explain Exercise 1 Take an A4 sheet of paper and a heay object (cricket ball, basketball, brick, book, etc). Predict what will
More informationVectors in R n. P. Danziger
1 Vectors in R n P. Danziger 1 Vectors The standard geometric definition of ector is as something which has direction and magnitude but not position. Since ectors hae no position we may place them whereer
More informationMath 144 Activity #9 Introduction to Vectors
144 p 1 Math 144 ctiity #9 Introduction to Vectors Often times you hear people use the words speed and elocity. Is there a difference between the two? If so, what is the difference? Discuss this with your
More informationA Geometric Review of Linear Algebra
A Geometric Reiew of Linear Algebra The following is a compact reiew of the primary concepts of linear algebra. I assume the reader is familiar with basic (i.e., high school) algebra and trigonometry.
More informationUnit 11: Vectors in the Plane
135 Unit 11: Vectors in the Plane Vectors in the Plane The term ector is used to indicate a quantity (such as force or elocity) that has both length and direction. For instance, suppose a particle moes
More informationCHAPTER 3 : VECTORS. Definition 3.1 A vector is a quantity that has both magnitude and direction.
EQT 101-Engineering Mathematics I Teaching Module CHAPTER 3 : VECTORS 3.1 Introduction Definition 3.1 A ector is a quantity that has both magnitude and direction. A ector is often represented by an arrow
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 informationBlue and purple vectors have same magnitude and direction so they are equal. Blue and green vectors have same direction but different magnitude.
A ector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the ector represents the magnitude and the arrow indicates the direction of the ector. Blue and
More informationRoberto s Notes on Linear Algebra Chapter 9: Orthogonality Section 2. Orthogonal matrices
Roberto s Notes on Linear Algebra Chapter 9: Orthogonality Section 2 Orthogonal matrices What you need to know already: What orthogonal and orthonormal bases for subspaces are. What you can learn here:
More informationLECTURE 1: INTRODUCTION, VECTORS, AND DOT PRODUCTS
LECTURE 1: INTRODUCTION, VECTORS, AND DOT PRODUCTS MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1. Introduction Firstly, I want to point out that I am here to help, so if you hae a question about a lecture
More informationAnnouncements Wednesday, September 05
Announcements Wednesday, September 05 WeBWorK 2.2, 2.3 due today at 11:59pm. The quiz on Friday coers through 2.3 (last week s material). My office is Skiles 244 and Rabinoffice hours are: Mondays, 12
More informationChapter 4: Techniques of Circuit Analysis
Chapter 4: Techniques of Circuit Analysis This chapter gies us many useful tools for soling and simplifying circuits. We saw a few simple tools in the last chapter (reduction of circuits ia series and
More informationThe Dot Product Pg. 377 # 6ace, 7bdf, 9, 11, 14 Pg. 385 # 2, 3, 4, 6bd, 7, 9b, 10, 14 Sept. 25
UNIT 2 - APPLICATIONS OF VECTORS Date Lesson TOPIC Homework Sept. 19 2.1 (11) 7.1 Vectors as Forces Pg. 362 # 2, 5a, 6, 8, 10 13, 16, 17 Sept. 21 2.2 (12) 7.2 Velocity as Vectors Pg. 369 # 2,3, 4, 6, 7,
More informationChapter 3. Systems of Linear Equations: Geometry
Chapter 3 Systems of Linear Equations: Geometry Motiation We ant to think about the algebra in linear algebra (systems of equations and their solution sets) in terms of geometry (points, lines, planes,
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 informationRoberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 4. Matrix products
Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 4 Matrix products What you need to know already: The dot product of vectors Basic matrix operations. Special types of matrices What you
More informationReview of Matrices and Vectors 1/45
Reiew of Matrices and Vectors /45 /45 Definition of Vector: A collection of comple or real numbers, generally put in a column [ ] T "! Transpose + + + b a b a b b a a " " " b a b a Definition of Vector
More information2) If a=<2,-1> and b=<3,2>, what is a b and what is the angle between the vectors?
CMCS427 Dot product reiew Computing the dot product The dot product can be computed ia a) Cosine rule a b = a b cos q b) Coordinate-wise a b = ax * bx + ay * by 1) If a b, a and b all equal 1, what s the
More informationPhysics 2A Chapter 3 - Motion in Two Dimensions Fall 2017
These notes are seen pages. A quick summary: Projectile motion is simply horizontal motion at constant elocity with ertical motion at constant acceleration. An object moing in a circular path experiences
More informationVIII - Geometric Vectors
MTHEMTIS 0-NY-05 Vectors and Matrices Martin Huard Fall 07 VIII - Geometric Vectors. Find all ectors in the following parallelepiped that are equialent to the gien ectors. E F H G a) b) c) d) E e) f) F
More informationAre the two relativistic theories compatible?
Are the two relatiistic theories compatible? F. Selleri Dipartimento di Fisica - Uniersità di Bari INFN - Sezione di Bari In a famous relatiistic argument ("clock paradox") a clock U 1 is at rest in an
More information4-vectors. Chapter Definition of 4-vectors
Chapter 12 4-ectors Copyright 2004 by Daid Morin, morin@physics.harard.edu We now come to a ery powerful concept in relatiity, namely that of 4-ectors. Although it is possible to derie eerything in special
More informationMath 425 Lecture 1: Vectors in R 3, R n
Math 425 Lecture 1: Vectors in R 3, R n Motiating Questions, Problems 1. Find the coordinates of a regular tetrahedron with center at the origin and sides of length 1. 2. What is the angle between the
More informationRoberto s Notes on Linear Algebra Chapter 11: Vector spaces Section 1. Vector space axioms
Roberto s Notes on Linear Algebra Chapter 11: Vector spaces Section 1 Vector space axioms What you need to know already: How Euclidean vectors work. What linear combinations are and why they are important.
More informationRoberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7. Inverse matrices
Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 7 Inverse matrices What you need to know already: How to add and multiply matrices. What elementary matrices are. What you can learn
More informationHYPOTHESIS TESTING SAMPLING DISTRIBUTION
Introduction to Statistics in Psychology PSY Professor Greg Francis Lecture 5 Hypothesis testing for two means Why do we let people die? HYPOTHESIS TESTING H : µ = a H a : µ 6= a H : = a H a : 6= a always
More informationEigenvalues and eigenvectors
Roberto s Notes on Linear Algebra Chapter 0: Eigenvalues and diagonalization Section Eigenvalues and eigenvectors What you need to know already: Basic properties of linear transformations. Linear systems
More informationUniformity of Theta- Assignment Hypothesis. Transit i ve s. Unaccusatives. Unergatives. Double object constructions. Previously... Bill lied.
Preiously... Uniformity of Theta- Assignment Hypothesis, daughter of P = Agent, daughter of P = Theme PP, daughter of! = Goal P called P Chris P P books gae P to PP Chris Unaccusaties Transit i e s The
More informationLecture #8-6 Waves and Sound 1. Mechanical Waves We have already considered simple harmonic motion, which is an example of periodic motion in time.
Lecture #8-6 Waes and Sound 1. Mechanical Waes We hae already considered simple harmonic motion, which is an example of periodic motion in time. The position of the body is changing with time as a sinusoidal
More information8.0 Definition and the concept of a vector:
Chapter 8: Vectors In this chapter, we will study: 80 Definition and the concept of a ector 81 Representation of ectors in two dimensions (2D) 82 Representation of ectors in three dimensions (3D) 83 Operations
More information(a)!! d = 17 m [W 63 S]!! d opposite. (b)!! d = 79 cm [E 56 N] = 79 cm [W 56 S] (c)!! d = 44 km [S 27 E] = 44 km [N 27 W] metres. 3.
Chapter Reiew, pages 90 95 Knowledge 1. (b). (d) 3. (b) 4. (a) 5. (b) 6. (c) 7. (c) 8. (a) 9. (a) 10. False. A diagram with a scale of 1 cm : 10 cm means that 1 cm on the diagram represents 10 cm in real
More informationUsing matrices to represent linear systems
Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 4 Using matrices to represent linear systems What you need to know already: What a linear system is. What elementary operations
More informationDO PHYSICS ONLINE. WEB activity: Use the web to find out more about: Aristotle, Copernicus, Kepler, Galileo and Newton.
DO PHYSICS ONLINE DISPLACEMENT VELOCITY ACCELERATION The objects that make up space are in motion, we moe, soccer balls moe, the Earth moes, electrons moe, - - -. Motion implies change. The study of the
More informationA summary of factoring methods
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 A summary of factoring methods What you need to know already: Basic algebra notation and facts. What you can learn here: What
More information6.3 Vectors in a Plane
6.3 Vectors in a Plane Plan: Represent ectors as directed line segments. Write the component form of ectors. Perform basic ector operations and represent ectors graphically. Find the direction angles of
More informationPurpose of the experiment
Centripetal Force PES 116 Adanced Physics Lab I Purpose of the experiment Determine the dependence of mass, radius and angular elocity on Centripetal force. FYI FYI Hershey s Kisses are called that because
More informationDoppler shifts in astronomy
7.4 Doppler shift 253 Diide the transformation (3.4) by as follows: = g 1 bck. (Lorentz transformation) (7.43) Eliminate in the right-hand term with (41) and then inoke (42) to yield = g (1 b cos u). (7.44)
More informationRapidly mixing Markov chains
Randomized and Approximation Algorithms Leonard Schulman Scribe: Rajneesh Hegde Georgia Tech, Fall 997 Dec., 997 Rapidly mixing Marko chains Goal: To sample from interesting probability distributions.
More informationChapter 11 Collision Theory
Chapter Collision Theory Introduction. Center o Mass Reerence Frame Consider two particles o masses m and m interacting ia some orce. Figure. Center o Mass o a system o two interacting particles Choose
More informationBrake applications and the remaining velocity Hans Humenberger University of Vienna, Faculty of mathematics
Hans Humenberger: rake applications and the remaining elocity 67 rake applications and the remaining elocity Hans Humenberger Uniersity of Vienna, Faculty of mathematics Abstract It is ery common when
More informationGeneral Lorentz Boost Transformations, Acting on Some Important Physical Quantities
General Lorentz Boost Transformations, Acting on Some Important Physical Quantities We are interested in transforming measurements made in a reference frame O into measurements of the same quantities as
More informationarxiv: v1 [math.co] 25 Apr 2016
On the zone complexity of a ertex Shira Zerbib arxi:604.07268 [math.co] 25 Apr 206 April 26, 206 Abstract Let L be a set of n lines in the real projectie plane in general position. We show that there exists
More informationHYPOTHESIS TESTING SAMPLING DISTRIBUTION. the sampling distribution for di erences of means is. 2 is known. normal if.
Introduction to Statistics in Psychology PSY Professor Greg Francis Lecture 5 Hypothesis testing for two sample case Why do we let people die? H : µ = a H a : µ 6= a H : = a H a : 6= a always compare one-sample
More informationRelativity in Classical Mechanics: Momentum, Energy and the Third Law
Relatiity in Classical Mechanics: Momentum, Energy and the Third Law Roberto Assumpção, PUC-Minas, Poços de Caldas- MG 37701-355, Brasil assumpcao@pucpcaldas.br Abstract Most of the logical objections
More informationAntiderivatives and indefinite integrals
Roberto s Notes on Integral Calculus Chapter : Indefinite integrals Section Antiderivatives and indefinite integrals What you need to know already: How to compute derivatives What you can learn here: What
More informationFrames of Reference, Energy and Momentum, with
Frames of Reference, Energy and Momentum, with Interactie Physics Purpose: In this lab we will use the Interactie Physics program to simulate elastic collisions in one and two dimensions, and with a ballistic
More information1 :: Mathematical notation
1 :: Mathematical notation x A means x is a member of the set A. A B means the set A is contained in the set B. {a 1,..., a n } means the set hose elements are a 1,..., a n. {x A : P } means the set of
More informationPatterns of Non-Simple Continued Fractions
Patterns of Non-Simple Continued Fractions Jesse Schmieg A final report written for the Uniersity of Minnesota Undergraduate Research Opportunities Program Adisor: Professor John Greene March 01 Contents
More informationSpecial types of matrices
Roberto s Notes on Linear Algebra Chapter 4: Matrix Algebra Section 2 Special types of matrices What you need to know already: What a matrix is. The basic terminology and notation used for matrices. What
More informationVectors and the Geometry of Space
Vectors and the Geometr of Space. Vectors in the Plane. Space Coordinates and Vectors in Space. The Dot Product of Two Vectors. The Cross Product of Two Vectors in Space.5 Lines and Planes in Space.6 Surfaces
More informationRoberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3. Diagonal matrices
Roberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3 Diagonal matrices What you need to know already: Basic definition, properties and operations of matrix. What you
More informationAETHER THEORY AND THE PRINCIPLE OF RELATIVITY 1
AEHER HEORY AND HE PRINIPLE OF RELAIVIY 1 Joseph Ley 4 Square Anatole France, 915 St Germain-lès-orbeil, France E. Mail: ley.joseph@orange.fr 14 Pages, 1 figure, Subj-lass General physics ABSRA his paper
More informationRoberto s Notes on Infinite Series Chapter 1: Sequences and series Section 4. Telescoping series. Clear as mud!
Roberto s Notes on Infinite Series Chapter : Sequences and series Section Telescoping series What you need to now already: The definition and basic properties of series. How to decompose a rational expression
More informationMon Apr dot product, length, orthogonality, projection onto the span of a single vector. Announcements: Warm-up Exercise:
Math 2270-004 Week 2 notes We will not necessarily finish the material from a gien day's notes on that day. We may also add or subtract some material as the week progresses, but these notes represent an
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 informationReversal in time order of interactive events: Collision of inclined rods
Reersal in time order of interactie eents: Collision of inclined rods Published in The European Journal of Physics Eur. J. Phys. 27 819-824 http://www.iop.org/ej/abstract/0143-0807/27/4/013 Chandru Iyer
More informationIntroduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles
Introduction to Thermodynamic Cycles Part 1 1 st Law of Thermodynamics and Gas Power Cycles by James Doane, PhD, PE Contents 1.0 Course Oeriew... 4.0 Basic Concepts of Thermodynamics... 4.1 Temperature
More information3.3 Operations With Vectors, Linear Combinations
Operations With Vectors, Linear Combinations Performance Criteria: (d) Mltiply ectors by scalars and add ectors, algebraically Find linear combinations of ectors algebraically (e) Illstrate the parallelogram
More informationLecture J. 10 Counting subgraphs Kirchhoff s Matrix-Tree Theorem.
Lecture J jacques@ucsd.edu Notation: Let [n] [n] := [n] 2. A weighted digraph is a function W : [n] 2 R. An arborescence is, loosely speaking, a digraph which consists in orienting eery edge of a rooted
More informationTransmission lines using a distributed equivalent circuit
Cambridge Uniersity Press 978-1-107-02600-1 - Transmission Lines Equialent Circuits, Electromagnetic Theory, and Photons Part 1 Transmission lines using a distributed equialent circuit in this web serice
More informationLESSON 4: INTEGRATION BY PARTS (I) MATH FALL 2018
LESSON 4: INTEGRATION BY PARTS (I) MATH 6 FALL 8 ELLEN WELD. Integration by Parts We introduce another method for ealuating integrals called integration by parts. The key is the following : () u d = u
More informationAnnouncements Wednesday, September 06. WeBWorK due today at 11:59pm. The quiz on Friday covers through Section 1.2 (last weeks material)
Announcements Wednesday, September 06 WeBWorK due today at 11:59pm. The quiz on Friday coers through Section 1.2 (last eeks material) Announcements Wednesday, September 06 Good references about applications(introductions
More informationChapter 1. The Postulates of the Special Theory of Relativity
Chapter 1 The Postulates of the Special Theory of Relatiity Imagine a railroad station with six tracks (Fig. 1.1): On track 1a a train has stopped, the train on track 1b is going to the east at a elocity
More informationSection 1.7. Linear Independence
Section 1.7 Linear Independence Motiation Sometimes the span of a set of ectors is smaller than you expect from the number of ectors. Span{, w} w Span{u,, w} w u This means you don t need so many ectors
More informationAnnouncements Wednesday, September 06
Announcements Wednesday, September 06 WeBWorK due on Wednesday at 11:59pm. The quiz on Friday coers through 1.2 (last eek s material). My office is Skiles 244 and my office hours are Monday, 1 3pm and
More informationNumber of solutions of a system
Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 7 Number of solutions of a system What you need to know already: How to solve a linear system by using Gauss- Jordan elimination.
More informationLesson 81: The Cross Product of Vectors
Lesson 8: The Cross Prodct of Vectors IBHL - SANTOWSKI In this lesson yo will learn how to find the cross prodct of two ectors how to find an orthogonal ector to a plane defined by two ectors how to find
More informationSPH4U UNIVERSITY PHYSICS
SPH4U UNIVERSITY PHYSICS REVOLUTIONS IN MODERN PHYSICS:... L (P.588-591) Special Relatiity Time dilation is only one of the consequences of Einstein s special theory of relatiity. Since reference frames
More informationPurpose of the experiment
Impulse and Momentum PES 116 Adanced Physics Lab I Purpose of the experiment Measure a cart s momentum change and compare to the impulse it receies. Compare aerage and peak forces in impulses. To put the
More informationKinematics on oblique axes
Bolina 1 Kinematics on oblique axes arxi:physics/01111951 [physics.ed-ph] 27 No 2001 Oscar Bolina Departamento de Física-Matemática Uniersidade de São Paulo Caixa Postal 66318 São Paulo 05315-970 Brasil
More informationChapter 2 Motion Along a Straight Line
Chapter Motion Along a Straight Line In this chapter we will study how objects moe along a straight line The following parameters will be defined: (1) Displacement () Aerage elocity (3) Aerage speed (4)
More informationFigure 1: Two projections of one vector onto another. Figure 2: Projecting a non-unit vector.
w w Figure 1: Two projections of one ector onto another. w Figure 2: Projecting a non-unit ector. 1 Intersecting a Ray with a Triangle But we don t care about whether the ray intersects the plane, we care
More informationAlgorithms and Data Structures 2014 Exercises and Solutions Week 14
lgorithms and ata tructures 0 xercises and s Week Linear programming. onsider the following linear program. maximize x y 0 x + y 7 x x 0 y 0 x + 3y Plot the feasible region and identify the optimal solution.
More informationMotors and Generators
Physics Motors and Generators New Reised Edition Brian Shadwick Contents Use the table of contents to record your progress through this book. As you complete each topic, write the date completed, then
More information6.4 VECTORS AND DOT PRODUCTS
458 Chapter 6 Additional Topics in Trigonometry 6.4 VECTORS AND DOT PRODUCTS What yo shold learn ind the dot prodct of two ectors and se the properties of the dot prodct. ind the angle between two ectors
More informationCofactors and Laplace s expansion theorem
Roberto s Notes on Linear Algebra Chapter 5: Determinants Section 3 Cofactors and Laplace s expansion theorem What you need to know already: What a determinant is. How to use Gauss-Jordan elimination to
More informationGeneral Physics I. Lecture 17: Moving Clocks and Sticks. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 17: Moing Clocks and Sticks Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ With Respect to What? The answer seems to be with respect to any inertial frame
More informationBc. Dominik Lachman. Bruhat-Tits buildings
MASTER THESIS Bc. Dominik Lachman Bruhat-Tits buildings Department of Algebra Superisor of the master thesis: Study programme: Study branch: Mgr. Vítězsla Kala, Ph.D. Mathematics Mathematical structure
More informationqwertyuiopasdfghjklzxcvbnmqwerty uiopasdfghjklzxcvbnmqwertyuiopasd fghjklzxcvbnmqwertyuiopasdfghjklzx cvbnmqwertyuiopasdfghjklzxcvbnmq
qwertyuiopasdfgjklzxcbnmqwerty uiopasdfgjklzxcbnmqwertyuiopasd fgjklzxcbnmqwertyuiopasdfgjklzx cbnmqwertyuiopasdfgjklzxcbnmq Projectile Motion Quick concepts regarding Projectile Motion wertyuiopasdfgjklzxcbnmqwertyui
More informationThe Inner Product (Many slides adapted from Octavia Camps and Amitabh Varshney) Much of material in Appendix A. Goals
The Inner Product (Many slides adapted from Octaia Camps and Amitabh Varshney) Much of material in Appendix A Goals Remember the inner product See that it represents distance in a specific direction. Use
More informationAbstract & Applied Linear Algebra (Chapters 1-2) James A. Bernhard University of Puget Sound
Abstract & Applied Linear Algebra (Chapters 1-2) James A. Bernhard University of Puget Sound Copyright 2018 by James A. Bernhard Contents 1 Vector spaces 3 1.1 Definitions and basic properties.................
More informationA vector in the plane is directed line segment. The directed line segment AB
Vector: A ector is a matrix that has only one row then we call the matrix a row ector or only one column then we call it a column ector. A row ector is of the form: a a a... A column ector is of the form:
More informationChapter 14 Thermal Physics: A Microscopic View
Chapter 14 Thermal Physics: A Microscopic View The main focus of this chapter is the application of some of the basic principles we learned earlier to thermal physics. This will gie us some important insights
More informationChapter 23 non calculus
Chapter 23 non calculus Magnetism CHAPTER 23 MAGNETISM In our discussion of Coulomb's law, we saw that electric forces are ery strong but in most circumstances tend to cancel. The strength of the forces
More informationBasic matrix operations
Roberto s Notes on Linear Algebra Chapter 4: Matrix algebra Section 3 Basic matrix operations What you need to know already: What a matrix is. he basic special types of matrices What you can learn here:
More informationarxiv: v2 [math.co] 12 Jul 2009
A Proof of the Molecular Conjecture Naoki Katoh and Shin-ichi Tanigawa arxi:0902.02362 [math.co] 12 Jul 2009 Department of Architecture and Architectural Engineering, Kyoto Uniersity, Kyoto Daigaku Katsura,
More informationLesson 3: Free fall, Vectors, Motion in a plane (sections )
Lesson 3: Free fall, Vectors, Motion in a plane (sections.6-3.5) Last time we looked at position s. time and acceleration s. time graphs. Since the instantaneous elocit is lim t 0 t the (instantaneous)
More informationLecture 3. (2) Last time: 3D space. The dot product. Dan Nichols January 30, 2018
Lectre 3 The dot prodct Dan Nichols nichols@math.mass.ed MATH 33, Spring 018 Uniersity of Massachsetts Janary 30, 018 () Last time: 3D space Right-hand rle, the three coordinate planes 3D coordinate system:
More informationALGEBRAIC TOPOLOGY IV. Definition 1.1. Let A, B be abelian groups. The set of homomorphisms ϕ: A B is denoted by
ALGEBRAIC TOPOLOGY IV DIRK SCHÜTZ 1. Cochain complexes and singular cohomology Definition 1.1. Let A, B be abelian groups. The set of homomorphisms ϕ: A B is denoted by Hom(A, B) = {ϕ: A B ϕ homomorphism}
More information27 ft 3 adequately describes the volume of a cube with side 3. ft F adequately describes the temperature of a person.
VECTORS The stud of ectors is closel related to the stud of such phsical properties as force, motion, elocit, and other related topics. Vectors allow us to model certain characteristics of these phenomena
More informationOrder of Operations. Real numbers
Order of Operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply and divide from left to right. 4. Add
More informationPhysics 240: Worksheet 24 Name:
() Cowboy Ryan is on the road again! Suppose that he is inside one of the many caerns that are found around the Whitehall area of Montana (which is also, by the way, close to Wheat Montana). He notices
More informationChapter 2. Linear Algebra. rather simple and learning them will eventually allow us to explain the strange results of
Chapter 2 Linear Algebra In this chapter, we study the formal structure that provides the background for quantum mechanics. The basic ideas of the mathematical machinery, linear algebra, are rather simple
More informationVector Basics, with Exercises
Math 230 Spring 09 Vector Basics, with Exercises This sheet is designed to follow the GeoGebra Introduction to Vectors. It includes a summary of some of the properties of vectors, as well as homework exercises.
More information