Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Vectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:"

Transcription

1 Vectors I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( ) 2, 3, (0,0). 5 The numers which mke up the vector re the vector s components. Here re some 3-dimensionl vectors: ( ) 1 (1,2, 17), 2, 17,π, (0,0,0). Since we usully use x, y, nd z s the coordinte vriles in 3 dimensions, vector s components re sometimes referred to s its x, y, nd z-components. For instnce, the vector (1, 2, 17) hs x-component 1, it hs y-component 2, nd it hs z-component 17. The set of 2-dimensionl rel-numer vectors is denoted R 2, just like the set of ordered pirs of rel numers. Likewise, the set of 3-dimensionl rel-numer vectors is denoted R 3. Geometriclly, vector is represented y n rrow. Here re some 2-dimensionl vectors: x A vector is commonly denoted y putting n rrow ove its symol, s in the picture ove. Here re some 3-dimensionl vectors: z x y The reltionship etween the lgeric nd geometric descriptions comes from the following fct: The vector from point P(,) to point Q(c,d) is given y PQ = (c,d ). In 3 dimensions, the vector from point P(,,c) to point Q(d,e,f) is PQ = (d,e,f c). Remrk. You ve proly lredy noticed the following hrmless confusion: (3, 2) cn denote the point (3,2) in the x-y-plne, or the 2-dimensionl rel vector (3,2). Notice tht the vector from the origin (0,0) 1

2 to the point (3,2) is the vector (3,2). 2 (3,2) (0,0) 3 So we cn usully regrd them s interchngele. When there s need to mke distinction, I will cll it out. Exmple. () Find the vector from P(3,4, 7) to Q( 2,2,5). () Find the vectors AB, BA, AC, nd CD for the points A(1,1), B(2,3), C( 2,0), nd D( 1,2). Sketch the vectors AB nd CD. () () PQ = ( 2 3,2 4,5 ( 7)) = ( 5, 2,12). AB = (2 1,3 1) = (1,2), BA = (1 2,1 3) = ( 1, 2), AC = ( 2 1,0 1) = ( 3, 1), CD = ( 1 ( 2),2 0) = (1,2). Notice tht BA = AB; this is true in generl. Here s sketch of the vectors AB nd CD: y B D A C x AB nd CD re oth (1,2); in the picture, you cn see tht the rrows which represent the vectors hve the sme length nd the sme direction. Geometriclly, two vectors (thought of s rrows) re equl if they hve the sme length nd point in the sme direction. 2

3 Exmple. In the picture elow, ssume the two lines re prllel. Which of the vectors, c, d is equl to the vector? is not equl to ; it hs the sme direction, ut not the sme length. c is not equl to ; it hs the sme length, ut the opposite direction. d is equl to, since it hs the sme length nd direction. Algericlly, two vectors re equl if their corresponding components re equl. Exmple. Find nd such tht (+2, ) = ( 8,7). Set the corresponding components equl nd solve for nd : + 2 = 8 = 7 3 = 15 / 3 3 = 5 Sustituting this into +2 = 8, I get 10 = 8, so = 2. The solution is = 2, = 5. The length of geometric vector is the length of the rrow tht represents it. The length of n lgeric vector is given y the distnce formul. If v = (,,c), the length of v is A vector with length 1 is clled unit vector. v = c 2. Exmple. () Find the length of (3,12, 4). ( 4 () Show tht 5, 3 ) is unit vector. 5 () (3,12, 4) = ( 4) 2 = 13. 3

4 () ( 4 5, 3 ) ( 4 ) 2 = ( ) = = 1. Algericlly, you dd or sutrct vectors y dding or sutrcting corresponding components: (,)+(c,d) = (+c,+d), (,) (c,d) = ( c, d). (Use n nlogous procedure to dd or sutrct 3-dimensionl vectors.) You cn t dd or sutrct vectors with different numers of components. For exmple, you cn t dd 2 dimensionl vector to 3 dimensionl vector. Algericlly, you multiply vector y numer y multiplying ech component y the numer: k (,) = (k,k). Vectors tht re multiples re sid to e prllel. Exmple. Compute: () (1,5)+(7,19). () (2, 3,8) (16,11,0). (c) 6 (5,3). (d) 2 (2, 1,2)+4 (1, 1,3). () () (c) (d) (1,5)+(7,19) = (1+7,5+19) = (8,24). (2, 3,8) (16,11,0) = (2 16, 3 11,8 0) = ( 14, 14,8). 6 (5,3) = (6 5,6 3) = (30,18). 2 (2, 1,2)+4 (1, 1,3) = (4, 2,4)+(4, 4,12) = (8, 6,16). Here re some properties of vector rithmetic. There is nothing surprising here. Proposition. Let u, v, nd w e vectors (in the sme spce) nd let k e rel numer. () (Associtivity) ( u+ v)+ w = u+( v + w). () (Commuttivity) u+ v = v + u. (c) (Zero vector) The vector 0 with ll-0 components stisfies 0+ u = u nd u+ 0 = u. (d) (Additive inverse) The dditive inverse u of u is the vector whose components re the negtives of the components of u. It stisfies u+( u) = 0. (e) (Distriutivity) k ( u+ v) = k u+k v. 4

5 Note: To sy tht the vectors re in the sme spce mens tht, for exmple, u, v, nd w re ll vectors in R 3. But ll of the results re true if u, v, nd w re vectors in R 100 (100-dimensionl Eucliden spce). Proof. The ide in ll these cses is to write the vectors in component form nd do the computtion. For exmple, here is proof of (c) in the cse tht u = (u 1,u 2,u 3 ) R u = (0,0,0)+(u 1,u 2,u 3 ) = (u 1,u 2,u 3 ) = u. u+ 0 = (u 1,u 2,u 3 )+(0,0,0) = (u 1,u 2,u 3 ) = u. Here is proof of (e). I ll consider the specil cse where u nd v re vectors in R 3. Thus, Then u = (u 1,u 2,u 3 ) nd v = (v 1,v 2,v 3 ). k ( u+ v) = k [(u 1,u 2,u 3 )+(v 1,v 2,v 3 )] = k (u 1 +v 1,u 2 +v 2,u 3 +v 3 ) = (k (u 1 +v 1 ),k (u 2 +v 2 ),k (u 3 +v 3 )) = (k u 1 +k v 1,k u 2 +k v 2,k u 3 +k v 3 ) = (k u 1,k u 2,k u 3 )+(k v 1,k v 2,k v 3 ) = k (u 1,u 2,u 3 )+k (v 1,v 2,v 3 ) = k u+k v The other prts re proved in similr fshion. There is n lternte nottion for vectors tht is often used in physics nd engineering. î, ĵ, nd ˆk re the unit vectors in the x, y, nd z directions: î = (1,0,0), ĵ = (0,1,0), ˆk = (0,0,1). Note tht For exmple, (,,c) = (1,0,0)+ (0,1,0)+c (0,0,1) = î+ĵ+cˆk. ( 3,6,10) = 3î+6ĵ+10ˆk. In 2 dimensions, (,) = î+ĵ. There is no î-ĵ-ˆk nottion for vectors with more thn 3 components. You operte with vectors using the î-ĵ-ˆk nottion in the ovious wys. For exmple, 2(3î 4ˆk)+(5î+7ĵ 11ˆk) = 11î+7ĵ 19ˆk. Geometriclly, multiplying vector y numer multiplies the length of the rrow y the numer. In ddition, if the numer is negtive, the rrow s direction is reversed: 2-3 5

6 You dd geometric vectors s shown elow. Move one of the vectors sy keeping its length nd direction unchnged so tht it strts t the end of the other vector. Since the copy hs the sme length nd direction s the originl, it s equl to. + Next, drw the vector which strts t the strting point of nd ends t the tip of. This vector is the sum +. The picture elow illustrtes why the geometric ddition rule follows from the lgeric ddition rule. It is oviously specil cse with two 2-dimensionl vectors with positive components, ut I think it mkes the result plusile. To dd severl vectors, move the vectors (keeping their lengths nd directions unchnged) so tht they re hed-to-til. In the second picture elow, I moved nd c. c c + + c c Finlly, drw vector from the strt of the first vector to the end of the lst vector. Tht vector is the sum in this cse, + + c. The picture elow shows how to sutrct one vector from nother in this cse, is the vector which goes from the tip of to the tip of. - 6

7 There re couple of wys to see this. First, if you interpret this s n ddition picture using the hed-to-til rule, it sys +( ) =. Alterntively, construct y flipping round, then dd to. This gives +( ) =. As the picture shows, it is the sme s the vector from the hed of to the hed of, ecuse the two vectors re opposite sides of prllelogrm. Exmple. Vectors nd re shown in the picture elow. Drw pictures of the vectors 2, 3 +2, nd 2. c 2017 y Bruce Ikeng 7

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

Lesson Notes: Week 40-Vectors

Lesson Notes: Week 40-Vectors Lesson Notes: Week 40-Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

On the diagram below the displacement is represented by the directed line segment OA.

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

Bases for Vector Spaces

Bases for Vector Spaces Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

More information

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 + Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

September 13 Homework Solutions

September 13 Homework Solutions College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are

More information

Linear Inequalities. Work Sheet 1

Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Coordinate geometry and vectors

Coordinate geometry and vectors MST124 Essentil mthemtics 1 Unit 5 Coordinte geometry nd vectors Contents Contents Introduction 4 1 Distnce 5 1.1 The distnce etween two points in the plne 5 1.2 Midpoints nd perpendiculr isectors 7 2

More information

Introduction to Group Theory

Introduction to Group Theory Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Quadratic Forms. Quadratic Forms

Quadratic Forms. Quadratic Forms Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

Lecture 3: Equivalence Relations

Lecture 3: Equivalence Relations Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Chapter 8.2: The Integral

Chapter 8.2: The Integral Chpter 8.: The Integrl You cn think of Clculus s doule-wide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in

More information

Lecture 08: Feb. 08, 2019

Lecture 08: Feb. 08, 2019 4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny

More information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

Boolean Algebra. Boolean Algebra

Boolean Algebra. Boolean Algebra Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with: - two inry opertions, commonly denoted y + nd, - unry opertion, usully denoted y ˉ or ~ or, - two elements usully clled zero nd

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the

More information

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω. Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

More information

CM10196 Topic 4: Functions and Relations

CM10196 Topic 4: Functions and Relations CM096 Topic 4: Functions nd Reltions Guy McCusker W. Functions nd reltions Perhps the most widely used notion in ll of mthemtics is tht of function. Informlly, function is n opertion which tkes n input

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

Nondeterminism and Nodeterministic Automata

Nondeterminism and Nodeterministic Automata Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information

Designing Information Devices and Systems I Discussion 8B

Designing Information Devices and Systems I Discussion 8B Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V

More information

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

More information

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements: Mtrices Elementry Mtrix Theory It is often desirble to use mtrix nottion to simplify complex mthemticl expressions. The simplifying mtrix nottion usully mkes the equtions much esier to hndle nd mnipulte.

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

Vectors. Introduction. Definition. The Two Components of a Vector. Vectors

Vectors. Introduction. Definition. The Two Components of a Vector. Vectors Vectors Introduction This pper covers generl description of vectors first (s cn e found in mthemtics ooks) nd will stry into the more prcticl res of grphics nd nimtion. Anyone working in grphics sujects

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS 3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

More information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

MTH 505: Number Theory Spring 2017

MTH 505: Number Theory Spring 2017 MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

Stage 11 Prompt Sheet

Stage 11 Prompt Sheet Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions

More information

Lecture 2e Orthogonal Complement (pages )

Lecture 2e Orthogonal Complement (pages ) Lecture 2e Orthogonl Complement (pges -) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

Infinite Geometric Series

Infinite Geometric Series Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

Section 3.2: Negative Exponents

Section 3.2: Negative Exponents Section 3.2: Negtive Exponents Objective: Simplify expressions with negtive exponents using the properties of exponents. There re few specil exponent properties tht del with exponents tht re not positive.

More information

1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.

1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and. Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

Network Analysis and Synthesis. Chapter 5 Two port networks

Network Analysis and Synthesis. Chapter 5 Two port networks Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on

More information

Designing finite automata II

Designing finite automata II Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of

More information

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4

Intermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4 Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

More information

EECS 141 Due 04/19/02, 5pm, in 558 Cory

EECS 141 Due 04/19/02, 5pm, in 558 Cory UIVERSITY OF CALIFORIA College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Lst modified on April 8, 2002 y Tufn Krlr (tufn@eecs.erkeley.edu) Jn M. Rey, Andrei Vldemirescu Homework

More information

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.

378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A. 378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),

More information

expression simply by forming an OR of the ANDs of all input variables for which the output is

expression simply by forming an OR of the ANDs of all input variables for which the output is 2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

3. Vectors. Home Page. Title Page. Page 2 of 37. Go Back. Full Screen. Close. Quit

3. Vectors. Home Page. Title Page. Page 2 of 37. Go Back. Full Screen. Close. Quit Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 37 3. Vectors Gols: To define vector components nd dd vectors. To introduce nd mnipulte unit vectors.

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Riemann Sums and Riemann Integrals

Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

Math 4310 Solutions to homework 1 Due 9/1/16

Math 4310 Solutions to homework 1 Due 9/1/16 Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

A B= ( ) because from A to B is 3 right, 2 down.

A B= ( ) because from A to B is 3 right, 2 down. 8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.

More information

M A T H F A L L CORRECTION. Algebra I 2 1 / 0 9 / U N I V E R S I T Y O F T O R O N T O

M A T H F A L L CORRECTION. Algebra I 2 1 / 0 9 / U N I V E R S I T Y O F T O R O N T O M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #1 CORRECTION Alger I 2 1 / 0 9 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O 1. Suppose nd re nonzero elements of field F. Using only the field xioms,

More information

y = f(x) This means that there must be a point, c, where the Figure 1

y = f(x) This means that there must be a point, c, where the Figure 1 Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

Introduction To Matrices MCV 4UI Assignment #1

Introduction To Matrices MCV 4UI Assignment #1 Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be

More information

Combinational Logic. Precedence. Quick Quiz 25/9/12. Schematics à Boolean Expression. 3 Representations of Logic Functions. Dr. Hayden So.

Combinational Logic. Precedence. Quick Quiz 25/9/12. Schematics à Boolean Expression. 3 Representations of Logic Functions. Dr. Hayden So. 5/9/ Comintionl Logic ENGG05 st Semester, 0 Dr. Hyden So Representtions of Logic Functions Recll tht ny complex logic function cn e expressed in wys: Truth Tle, Boolen Expression, Schemtics Only Truth

More information

Lecture 2: Fields, Formally

Lecture 2: Fields, Formally Mth 08 Lecture 2: Fields, Formlly Professor: Pdric Brtlett Week UCSB 203 In our first lecture, we studied R, the rel numbers. In prticulr, we exmined how the rel numbers intercted with the opertions of

More information

A study of Pythagoras Theorem

A study of Pythagoras Theorem CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

MAS 4156 Lecture Notes Differential Forms

MAS 4156 Lecture Notes Differential Forms MAS 4156 Lecture Notes Differentil Forms Definitions Differentil forms re objects tht re defined on mnifolds. For this clss, the only mnifold we will put forms on is R 3. The full definition is: Definition:

More information

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17 EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

More information

ODE: Existence and Uniqueness of a Solution

ODE: Existence and Uniqueness of a Solution Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.) CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

More information