Introduction to vectors


 Delilah Lawson
 1 years ago
 Views:
Transcription
1 Introduction to vectors mctyintrovector Avectorisquntityththsothmgnitude(orsize)nddirection. Bothofthese propertiesmustegiveninordertospecifyvectorcompletely.inthisunitwedescriehowto writedownvectors,howtoddndsutrctthem,ndhowtousethemingeometry. In order to mster the techniques explined here it is vitl tht you undertke plenty of prctice exercises so tht they ecome second nture. Afterredingthistext,nd/orviewingthevideotutorilonthistopic,youshouldeleto: distinguish etween vector nd sclr; understnd how to dd nd sutrct vectors; knowwhenonevectorismultipleofnother; use vectors to solve simple prolems in geometry. Contents 1. Introduction 2 2. Representing vector quntities 2 3. Position vectors 3 4. Some nottion for vectors 3 5. Adding two vectors 4 6. Sutrcting two vectors 5 7. Addingvectortoitself 5 8. Vectors of unit length 6 9. Using vectors in geometry c mthcentre 2009
2 1. Introduction Vector quntities re extremely useful in physics. The importnt chrcteristic of vector quntityisthtithsothmgnitude(orsize)nddirection. Bothofthesepropertiesmuste given in order to specify vector completely. Anexmpleofvectorquntityisdisplcement.Thistellushowfrwywerefromfixed point,nditlsotellsusourdirectionreltivetothtpoint. P O Another exmple of vector quntity is velocity. This is speed, in prticulr direction. An exmpleofvelocitymighte60mphduenorth. Aquntitywithmgnitudelone,utnodirection,isnotvector. Itisclledsclrinsted. Oneexmpleofsclrisdistnce.Thistellsushowfrwerefromfixedpoint,utdoesnot giveusnyinformtionoutthedirection.anotherexmpleofsclrquntityisthemssof n oject. Key Point A vector hs oth mgnitude nd direction, nd oth these properties must e given in order tospecifyit.aquntitywithmgnitudeutnodirectionisclledsclr. 2. Representing vector quntities Wecnrepresentvectorylinesegment.Thisdigrmshowstwovectors. B A Wehveusedsmllrrowtoindictethtthefirstvectorispointingfrom Ato B.Avector pointing from B to A would e going in the opposite direction. Sometimeswerepresentvectorwithsmlllettersuchs,inoldtypefce. Thisis common in textooks, ut it is inconvenient in hndwriting. In writing, we normlly put r underneth,orsometimesontopof,theletter: or.inspeech,wecllthisthevector r. 2 c mthcentre 2009
3 3. Position vectors Sometimesvectorsrereferredtofixedpoint,norigin. Suchvectorisclledposition vector. Sowemightrefertothepositionvectorofpoint P withrespecttonorigin O. In writing, might put OP for this vector. Alterntively, we could write it s r. These two expressions refer to the sme vector. P O r 4. Some nottion for vectors Whtdoesitmenif,fortwovectors, =?Thismensfirstthtthelengthof equlsthe lengthof,sothtthetwovectorshvethesmemgnitude.butitlsomenstht nd reinthesmedirection.howcnwewritethisdownmoresuccinctly? Iftwovectorsre inthesmedirection,thentheyreprllel.wewritethisdowns //. Forlength,ifwehvevector AB,wecnwriteitslengths ABwithoutther.Alterntively, wecnwriteits AB.Thetwoverticllinesgiveusthemodulus,orsizeof,thevector.Ifwe hvevectorwrittens,wecnwriteitslengthseither withtwoverticllines,ors inordinrytype(orwithoutther).thisiswhyitisveryimportnttokeeptotheconvention ththseendoptedinordertodistinguishetweenvectornditslength. Key Point Thelengthofvector ABiswrittens ABor AB, ndthelengthofvector iswrittens (inordinrytype,orwithoutther)ors. Iftwovectors nd reprllel,wewrite // 3 c mthcentre 2009
4 5. Adding two vectors Oneofthethingswecndowithvectorsistoddthemtogether. Weshllstrtydding twovectorstogether. Oncewehvedonetht,wecnddnynumerofvectorstogethery ddingthefirsttwo,thenddingtheresulttothethird,ndsoon. Inordertoddtwovectors,wethinkofthemsdisplcements. Wecrryoutthefirstdisplcement, nd then the second. So the second displcement must strt where the first one finishes. + Thesumofthevectors, + (ortheresultnt,sitissometimesclled)iswhtwegetwhen wejoinupthetringle.thisisclledthetringlelwforddingvectors. There is nother wy of dding two vectors. Insted of mking the second vector strt where thefirstonefinishes,wemkethemothstrttthesmeplce,ndcompleteprllelogrm. This is clled the prllelogrm lw for dding vectors. It gives the sme result s the tringle lw,ecuseoneofthepropertiesofprllelogrmisthtoppositesidesreequlndinthe smedirection,sotht isrepetedtthetopoftheprllelogrm. + Key Point Wecnddtwovectors nd ymking strtwhere finishes, ndcompletingthe tringle.alterntively,wecnmke nd strttthesmeplce,ndtkethedigonlof the prllelogrm. 4 c mthcentre 2009
5 6. Sutrcting two vectors Whtis?Wethinkofthiss + ( ),ndthenweskwht mightmen.thiswill evectorequlinmgnitudeto,utinthereversedirection. Nowwecnsutrcttwovectors.Sutrcting from willethesmesdding to. Key Point mens + ( ) 7. Adding vector to itself Wht hppens when you dd vector to itself, perhps severl times? We write, for exmple, + + = 3. Inthesmewy,wewouldwrite n = } + {{... + }. ncopies 5 c mthcentre 2009
6 Key Point Avector nisinthesmedirectionsthevector,ut ntimesslong. 8. Vectors of unit length Thereisonemorepieceofnottionweshllusewhenwritingvectors.If isnyvector,weshll write âtorepresentunitvectorinthedirectionof.auntvectorisvectorwhoselengthis 1,sotht â = 1. Thisnottiongivesusnotherwyofwritingthevector :wecnwriteits â,sothtitis the length multiplied y the unit vector â. Key Point Aunitvectorinthedirectionofthevector iswrittens â,sotht = â. 9. Using vectors in geometry Exmple There is useful theorem in geometry clled the midpoint theorem. In this theorem, we tke twopoints And B,definedwithrespecttonorigin O.Letuswrite forthepositionvector of A,nd forthepositionvectorof B.Wecnjoin And Bwithline,togivetringle. Nowtkethemidpoint Moftheline OA,ndthemidpoint Noftheline OB,ndjoin Mto Nwithline. Cnwesynythingoutthereltionshipetweentheline MN ndtheline AB? 6 c mthcentre 2009
7 M A O N B Wecnnswerthisveryesilywithvectors. Wecnwritethevectorforthelinesegment AB s AO + OB.Now AOisthereverseofthevector,soitis. And OBisthesmesthe vector. Therefore AB = AO + OB = ( ) + =. Whtout MN?Followingthesmeresoning,thisis MO +ON.Butwhtis MO?Thisis vectorhlfthelengthof AO,ndinthesmedirection,soitmuste 1 2 ( ).Inthesmewy, ONisinthesmedirections OB,utishlfthelength,soitmuste 1 2.Therefore MN = MO + ON = 1 2 ( ) = 1 ( ). 2 Nowwecncompre ABnd MN. Fromourclcultion,wecnseetht MNis 1 AB. So, 2 sthisisvectoreqution,ittellsustwothings. First,ittellsusoutmgnitude,sotht MN = 1 AB.Also,ittellsustht MNnd ABmusteinthesmedirection,sotht MN//AB. 2 Thisisclledthemidpointtheoremfortringle. Itsttesthtifyoujointhemidpointsof twosidesoftringlethentheresultinglineisequltohlfofthethirdsideofthetringle,nd isprlleltoit. Exmple Wecnpplythemidpointtheoremtoqudrilterl,orindeedtonyfourpointsinspce,to giveninterestinggeometriclresult. Weshllcllthefourpoints A, B, Cnd D. Weshll lsogivelelstothemidpointsofthefoursides:weshllcllthemidpoints P, Q, Rnd S. Nowletusjointhefourmidpoints,tomkenewshpe PQRS.Whtkindofshpeisthis? B Q C R P D A S 7 c mthcentre 2009
8 Wecnidentifytheshpeyjoiningthepoints And C. Ifwepplythemidpointtheoremtotringle ABC,weseetht PQ = 1 2 AC. Butifwepplythemidpointtheoremtothetringle ADC,welsoseetht RS = 1 2 AC. If we comine these two equtions, we then otin PQ = RS. Nowthisisvectoreqution,ndsoittellsustwothings. First,ittellsusthtthelengthof PQisthesmesthelengthof RS. Andsecondly,ittellsusthtthedirectionof PQisthe smesthedirectionof RS,sotht PQnd RSreprllel. Buthvingtwoprllelsidesof equllengthispropertywhichdefinesprllelogrm,ndsotheshpe PQRSmuste prllelogrm. Exmple Weshllnowusevectorstoproveonemoretheorem. Tketwopoints And B,hvingpositionvectors, withrespecttonorigin O. Drwthe line AB,ndtkepoint P onthtlinewhichdividesitinthertioof mto n. Whtisthe positionvectorof Pwithrespectto O?. A m P r n B O Wecnusethesmemethodthtweusedefore.Weknowtht ndwelsoknowtht OA =.Butwhtis AP? OP = OA + AP, (1) Now APisinthesmedirections AB,ndtheirlengthsreinthertioof mto m + n.so Welsoknowtht AP = m AB. (2) m + n AB = AO + OB =. 8 c mthcentre 2009
9 Now we cn put these three sttements together, replcing AP in eqution(1) y using eqution(2),ndreplcing ABineqution(2)yusingtheeqution(3),sothteverythingwille writtenintermsof nd.thisgivesus OP = + m ( ). m + n Putting ll this over common denomintor then gives OP = (m + n) + m( ). m + n Ifweexpndtherckets,theterm mwillcncelwiththeterm m( ),ndsofinllywehve OP = n + m m + n. Thisformulgivesuswyofclcultingthepositionvectorofthepoint P.Forinstnce,if m nd nwereoth 1then Pwouldethemidpointof AB. Thepositionvectorofthemidpoint woulde ( + )/2.Asnotherexmple,if m = 2nd n = 1,sotht Pwstwothirdsofthe wylongtheline,thenthepositionvectorof Pwoulde ( + 2)/3. Exercises 1.Thevector isshownelow. Sketchthevectors 2, 3, 1 nd In OAB, OA = nd OB =.Intermsof nd, () Wht is AB? () Wht is BA? (c) Whtis OP,where Pisthemidpointof AB? (d) Whtis AP? (e) Whtis BP? (f) Whtis OQ,where Qdivides ABinthertio2:3? 3.Whtismentyunitvector? 4.If eisunitvector,whtisthelengthof 3e? 5.In ABC, AB =, BC =, CA = c.whtis + + c? 9 c mthcentre 2009
10 Answers () () (c) 1 1 (e) ( ) (f) Avectorwithlength ( + )(d) 1( ) c mthcentre 2009
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 informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors 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 informationChapter 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 informationAnalytically, vectors will be represented by lowercase boldface 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 informationCalculus 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 informationSection 3.1: Exponent Properties
Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted
More informationCoordinate 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 informationHow 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 information4 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 informationChapter 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 information5: 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 informationset 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 informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationDeterminants Chapter 3
Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More information1.2 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 fillinlnksprolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers
More informationFarey 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 informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationMA123, 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. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd AmGm inequlity 2. Elementry inequlities......................
More informationdy ky, dt where proportionality constant k may be positive or negative
Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationCS12N: 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 informationHomework 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 informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 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 information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More information4.1. Probability Density Functions
STT 1 4.14. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile  vers  discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationSimilarity and Congruence
Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles
More informationVECTOR ALGEBRA. Chapter Introduction Some Basic Concepts
44 Chpter 0 VECTOR ALGEBRA In most sciences one genertion ters down wht nother hs built nd wht one hs estblished nother undoes In Mthemtics lone ech genertion builds new story to the old structure HERMAN
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationWhat else can you do?
Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright
More informationFORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81
FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first
More informationImproper 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 informationThis chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationProject 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 informationMath 017. Materials With Exercises
Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationTriangles The following examples explore aspects of triangles:
Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x  4 +x xmple 3: ltitude of the
More information1 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 information8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationHomework Solution  Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution  et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte nonfinl.
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if dbc0. The expression dbc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
More informationSCHOOL 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( ) Straight line graphs, Mixed Exercise 5. 2 b The equation of the line is: 1 a Gradient m= 5. The equation of the line is: y y = m x x = 12.
Stright line grphs, Mied Eercise Grdient m ( y ),,, The eqution of the line is: y m( ) ( ) + y + Sustitute (k, ) into y + k + k k Multiply ech side y : k k The grdient of AB is: y y So: ( k ) 8 k k 8 k
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationHomework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)
More information3 Regular expressions
3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll
More informationContinuous 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 informationMATH 423 Linear Algebra II Lecture 28: Inner product spaces.
MATH 423 Liner Algebr II Lecture 28: Inner product spces. Norm The notion of norm generlizes the notion of length of vector in R 3. Definition. Let V be vector spce over F, where F = R or C. A function
More informationUnit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use
More informationCHAPTER 6 Introduction to Vectors
CHAPTER 6 Introduction to Vectors Review of Prerequisite Skills, p. 73 "3 ".. e. "3. "3 d. f.. Find BC using the Pthgoren theorem, AC AB BC. BC AC AB 6 64 BC 8 Net, use the rtio tn A opposite tn A BC djcent.
More informationESCI 241 Meteorology Lesson 0 Math and Physics Review
UNITS ESCI 41 Meteorolog Lesson 0 Mth nd Phsics Review A numer is meningless unless it is ccompnied unit telling wht the numer represents. The stndrd unit sstem used interntionll scientists is known s
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationMATH STUDENT BOOK. 10th Grade Unit 5
MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY
More informationLecture 2 : Propositions DRAFT
CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte
More informationWaveguide Guide: A and V. Ross L. Spencer
Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More informationIntroduction to Electrical & Electronic Engineering ENGG1203
Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 278 Dr. Hden KwokH So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll
More informationPreview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms
Preview Greed Algorithms Greed Algorithms Coin Chnge Huffmn Code Greed lgorithms end to e simple nd strightforwrd. Are often used to solve optimiztion prolems. Alws mke the choice tht looks est t the moment,
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder
More informationCS 311 Homework 3 due 16:30, Thursday, 14 th October 2010
CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w
More informationCS 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 informationThe solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the evectors nd evlues
More informationParallel Projection Theorem (Midpoint Connector Theorem):
rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length onehlf the third side. onversely, If line isects
More informationBasics of Olympiad Inequalities. Samin Riasat
Bsics of Olympid Inequlities Smin Rist ii Introduction The im of this note is to cquint students, who wnt to prticipte in mthemticl Olympids, to Olympid level inequlities from the sics Inequlities re used
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( 5, 9 ) ) (,
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More information42nd International Mathematical Olympiad
nd Interntionl Mthemticl Olympid Wshington, DC, United Sttes of Americ July 8 9, 001 Problems Ech problem is worth seven points. Problem 1 Let ABC be n cutengled tringle with circumcentre O. Let P on
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More information1 From NFA to regular expression
Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work
More informationLecture 7 notes Nodal Analysis
Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions
More informationGRADE 4. Division WORKSHEETS
GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of
More informationRegular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15
Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll
More informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationONLINE PAGE PROOFS. Antidifferentiation and introduction to integral calculus
Antidifferentition nd introduction to integrl clculus. Kick off with CAS. Antiderivtives. Antiderivtive functions nd grphs. Applictions of ntidifferentition.5 The definite integrl.6 Review . Kick off
More informationDATABASTEKNIK  1DL116
DATABASTEKNIK  DL6 Spring 004 An introductury course on dtse systems http://user.it.uu.se/~udl/dtvt004/ Kjell Orsorn Uppsl Dtse Lortory Deprtment of Informtion Technology, Uppsl University, Uppsl, Sweden
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More informationQuadratic Equations. Brahmagupta gave. Solving of quadratic equations in general form is often credited to ancient Indian mathematicians.
9 Qudrtic Equtions Qudrtic epression nd qudrtic eqution Pure nd dfected qudrtic equtions Solution of qudrtic eqution y * Fctoristion method * Completing the squre method * Formul method * Grphicl method
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationRegular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene*
Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationResources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations
Introduction: Binding Prt of 4lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More information3.1 Review of Sine, Cosine and Tangent for Right Angles
Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More information