1.3 SCALARS AND VECTORS


 Warren Rice
 1 years ago
 Views:
Transcription
1 Bridge Course Phy I PUC 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 the Sun is finite. The study of speed of light involves the distne trveled y the ry of light nd time onsumed. ny thing tht is mesurle is termed s quntity. The quntities tht ome ross in physis is referred to s physil quntity. Exmple: Mss, length, time, temperture, et., Whenever we mesure physil quntity, the mesured vlue is lwys numer. This numer mkes sense only when the relevnt unit is speified. Thus, the result of mesurement hs numeril vlue nd unit of mesure. For exmple, the mss of ody is 3 kg. Here quntity hving numeril vlue 3 nd the unit of mesure kg re used. The numeril vlue together with the unit is lled the mgnitude. To desrie ertin physil quntities only mgnitude is required. prt from the mss of ody, distne to ny ple, time, temperture, height, the numer of osilltions of pendulum nd the numer of ooks in g re some exmples of suh numers. They hve no diretion ssoited with them. Quntities whih require only mgnitude for their omplete speifitions nd hving no diretion ssoited with them re lled slr quntities. To desrie ertin physil quntities like displement long with the mgnitude, the diretion is essentil. Consider ody moving from X to Y. XY is the displement. On the ontrry, if the ody moves from Y to X, the displement is YX. Quntities whih require oth mgnitude nd diretion for their omplete speifition re lled vetors. Exmple for vetor quntities: momentum, fore, torque, mgneti field et., Note: The physil quntity like eletri urrent possesses oth the mgnitude nd diretion, still they re not vetors, nd similrly ny form of energy is slr. Representtion of vetor: vetor n e onveniently represented y stright line with n rrow hed. The length of the vetor represents its mgnitude nd the rrow hed indites its diretion. Steps involved representing vetor: 1. By hoosing proper sle, drw line whose length is proportionl to the mgnitude of the vetor. 2. By following the stndrd onvention to show diretion, indite the diretion of the vetor y mrking n rrow hed t one end of the line. Exmple: 1) To represent the displement of ody long xxis. N Sle: 1 m = 5 km W E 0 35 km S Fig 1 Grphil representtion of vetor The vetor represented y the direted line segment O in fig (1) is denoted y O (to e red s vetor O) or simple nottion s ( to e red s vetor ). For vetoro, O is the initil point nd is the terminl point.
2 The mgnitude of O (or ) is denoted y O, (red s modulus of vetor O) or ( or simply O or ) nd is lwys positive. Position vetor: To lote the position of ny point P in Y P(x,y) plne or spe, generlly fixed point of referene lled the origin O is tken. The vetor OP is lled the position O X vetor of P with respet to O s shown in fig (2). Fig 2 Note: i) Given point P, there is one nd only one position vetor for the point with respet to the origin O. ii) Position vetor of point P hnges if the position of the origin O is hnged. Kinds of vetors: 1) Unit vetor: vetor hving unit mgnitude is lled unit vetor. If is vetor hving mgnitude 0, then is unit vetor hving the sme diretion s r. It is represented s (red s p). = 1. Thus = or = =. The unit vetors long the x, y nd zxis re usully denoted s i, j nd k respetively. 2) Zero vetor or Null vetor: vetor hving zero mgnitude is lled Null vetor or Zero vetor. It is represented s O. Note: ) Zero vetor hs no speifi diretion. ) The position vetor of origin is zero vetor. ) Zero vetors re only of mthemtil importne. 3) Equl vetors: Vetors re sid to e equl if oth vetors hve sme mgnitude nd diretion. Fig (3) = 4) Prllel vetors (Like vetors): Vetors re sid to e prllel if they hve the sme diretions. Fig (4) The vetors nd represent prllel vetors. Note: Two equl vetors re lwys prllel ut, two prllel vetors my or my not e equl vetors. Bridge Course Phy I PUC 25
3 5) nti prllel vetors (Unlike vetors): Vetors re sid to e nti prllel if they ts in opposite diretion. Fig (5) The vetors nd re nti prllel vetors. 6) Negtive vetor : The negtive vetor of ny vetor is vetor hving equl mgnitude ut ts in opposite diretion. Fig (6) =  OR =  7) Conurrent vetors (Co initil vetors ): vetors hving the sme initil point re lled onurrent vetors or o initil vetors. O, nd re onurrent t point O. Fig (7) 8) Coplnr vetors: The vetors in the sme plne re lled oplnr vetors. O Fig (8) The vetor nd re oplnr vetors Fig (8) The vetors nd re onurrent oplnr vetors. 9) Orthogonl vetors: Two vetors re sid to e orthogonl to one nother if the ngle etween them is 90. The vetor nd re orthogonl to one nother. 0 Fig (9) VECTOR LGEBR: ddition of vetors: The ddition of slrs involves only the ddition of their mgnitudes. But, when vetor is dded with nother vetor we hve to onsider their diretion lso. vetor n e dded with nother vetor provided oth the vetors represents the sme physil quntity. For exmple, the ddition of vetor representing displement of ody with nother vetor representing veloity of the ody is meningless. Bridge Course Phy I PUC 26
4 METHODS OF VECTOR DDITION: I.Tringle method of vetor ddition OR Til to tip method of vetor ddition: C Illustrtion: + = + = B Fig (10) Explntion: To dd with, trnslte, y drwing prllel to itself so tht the origin or initil point of is t the tip of vetor. nd re two vetors represented y two sides of tringle tken in the sme sense (diretion). The vetor sum of nd (lso lled resultnt of nd ) is represented y the third side of the tringle tken in opposite sense( diretion). Sttement: Tringle lw of vetor ddition sttes tht if two vetors n e represented in mgnitude nd diretion y two sides of tringle tken in the sme order, then their resultnt is represented ompletely y the third side of the tringle tken in opposite order. II. Prllelogrm method of vetor ddition: To dd two vetors pled with ommon initil point, the prllelogrm method of vetor is used. Illustrtion: B C = θ θ θ O O O α + = + = Fig (11) Fig (11) Fig (11) Explntion: To dd vetor with inlined t n ngleθ, drw equl vetor of t the tip of. By lw of tringle method of vetor ddition = + (fig11). Repet the proess, y drwing equl vetor of t the tip of (fig11). gin y lw of tringle method of vetor ddition = +. Note tht + = +, tht is vetor ddition follows ommuttive rule., the digonl of the ompleted prllelogrm represents the vetor sum of nd ompletely oth in mgnitude nd diretion. Sttement of prllelogrm lw of vetor ddition: It sttes tht if two vetors ting t point n e represented oth in mgnitude nd diretion y the two djent sides of prllelogrm drwn from tht point, the resultnt is represented ompletely y the digonl of the prllelogrm pssing through tht point. In fig(11), if nd re two vetors nd θ is the ngle etween them, then their vetor sum is represented y the digonl. Bridge Course Phy I PUC 27
5 It n e shown tht the mgnitude of If α is the ngle mde y the diretion of with tht of, then sinθ tnα = + osθ is, = osθ Note: It is ommon mistke to drw the sum vetor s the digonl running etween the tips of the two vetors s shown in fig (12). + = Fig (12) Think: In ft, the digonl represents the differene etween the two vetors, not their sum! Note: 1) The dvntge of the prllelogrm method is tht one n get oth the sum nd the differene of two vetors if one knows how to identify the pproprite diretions. 2) The resultnt of two vetors does not depend on the order in whih the vetors re dded. This ft leds to ) Commuttive lw of vetor ddition: + = + ) ssoitive lw of vetor ddition: If, nd re three vetor,then +( + )= ( + ) + III. Lw of polygon of vetor ddition: Sttement: It sttes tht if numer of vetors re represented oth in mgnitude nd diretion y the sides of polygon tken in the sme order, then their sum (resultnt) is represented oth in mgnitude nd diretion y the losing side of the polygon tken in the opposite order. C Let the vetors,, nd d represent the sides d O, B, BC nd CD of the polygon OBCDO. D B Their resultnt R is represented y the losing side OD of the polygon tken in the opposite order. R O Fig (13) Sutrtion of two vetors: Sutrtion of one vetor from nother vetor n e relized using the definition of negtive of vetor s follows.  = + Fig (14) Bridge Course Phy I PUC 28
6 Tringle method:  = Fig (15) Prllelogrm method:  = Fig (16) Note: 1) Sutrtion of one vetor with nother vetor is regrded s the ddition of one vetor with negtive of nother vetor. 2) The knowledge of sutrtion of vetors is useful in understnding the onept of reltive veloity. RESOLUTION OF VECTORS: Resolution of vetor mens the proess of splitting of vetor into omponents. If vetor is resolved into two omponents long the two mutully perpendiulr diretions, they re lled retngulr omponents. Consider vetor R represented y OC oth in mgnitude nd diretion s shown in fig (17). Drw OX nd OY whih re mutully perpendiulr to eh other from O. Y Let the vetor R mkes n ngle θ with Xxis. Drop B C Perpendiulrs from the tip of vetor C to X nd Y xes. O = P is the omponent of R long Xxis nd Q R is lled horizontl omponent of R. OB = Q is the θ omponent of R long Yxis nd is lled O P X the vertil omponent of R. Fig (17) From fig (17), R = P + Q = i P + j Q Where i nd j re the unit vetors ting long X nd Y xes respetively. From fig(17), the mgnitude of O is O = osθ O = OC osθ P = Rosθ OC The mgnitude of OB is C = sinθ OC C = OC sinθ Q = R sinθ Thus, R = (R osθ) i + (Rsinθ) j C Q lso, = tnθ or tnθ =, where θ gives the resultnt diretion. O P From geometry of the figure, it n e shown tht, R 2 =P 2 + Q 2 ************* Bridge Course Phy I PUC 29
7 QUESTIONS: 1. Wht is slr? 2. Identify whether the following quntities re slrs or vetors? (i) Mss (ii) weight (iii)speed (iv)veloity (v)energy (vi)work (vii)fore (ix)temperture (x)pressure (xi)ngulr momentum (xii)wvelength. 3. Wht is vetor? 4. Find the mgnitude of 4î + 3 ĵ 5. Two vetors P nd Q t t n ngle 60 0 with eh other. If P = 20 units nd Q = 8 units find the mgnitude of resultnt R. 6. If = 6 î + 3 ĵ nd B = 3 î + 2 ĵ find + B nd B. 7. If = 3 î 2 ĵ + kˆ, find the unit vetor C 8. When will e P + Q = O 9. vetor of mgnitude 10 units mkes n ngle of 30 0 with the Xxis. Find its X nd Y omponents. 10. For wht ngle the mgnitude of X nd Y omponents of vetor eome equl? Bridge Course Phy I PUC 30
8 NSWERS : 1) physil quntity whih requires only mgnitude for their omplete desription re lled slrs. 2) SCLRS: mss, speed, energy, work, temperture nd pressure. VECTORS: Weight, veloity, fore, ngulr momentum nd wvelength. 3) Physil quntity whih requires oth the mgnitude nd diretion for their omplete desription re lled vetors. 4) Given vetor is 4î + 3 ĵ The mgnitude of the vetor, r = = 4 3 = 25 units. 5) Given: P = 20 units nd Q = 8 units nd θ =60 0 R =? Mgnitude of the resultnt vetor R is 2 2 R = P + Q + 2 P Q osθ 2 2 o = (20)(8) os 60 = (20)(8)( ) = units 6) Given: 6 î + 3 ĵ nd B = 3 î + 2 ĵ i) + B = (6 î + 3 ĵ ) + (3 î + 2 ĵ ) = 9 î + 5 ĵ ii ) B = (6 î + 3 ĵ )  (3 î + 2 ĵ ) =3 î + ĵ 7) Given: = 3 î 2 ĵ + kˆ then Â =? We know tht, Â = 3ˆ i 2 ˆj + kˆ Â = ( 2) + 1 = 3i ˆ 2 ˆj + kˆ 14 8) P + Q = O if Q = P i.e., the mgnitude of Q is equl to the mgnitude of P nd ts in opposite diretion. (Note: Two equl vetors ting in opposite diretion nel eh other) 9) Given: θ =30 0 with Xxis Xomponent=10os30 0 = 10(0.8660) =8.660 units Yomponent =10sin30 0 = 10(0.5000) =5.000 units. 10) Let the vetor e R Its Xomponent is Ros nd Y omponent is Rsin Given: Ros = Rsin os = sin this is possile only when θ =45 0 Bridge Course Phy I PUC 31
Analytically, 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 informationSTRAND 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 informationm A 1 1 A ! and AC 6
REVIEW SET A Using sle of m represents units, sketh vetor to represent: NONCALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in northesterl diretion. Simplif:
More informationVectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:
hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points
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 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 informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The oordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
COORDINTE GEOMETR II I Qudrnt Qudrnt (.+) (++) X X    0  III IV Qudrnt  Qudrnt ()  (+) Region CRTESIN COORDINTE SSTEM : Retngulr Coordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
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 informationH (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More information2.1 ANGLES AND THEIR MEASURE. y I
.1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the
More informationStandard Trigonometric Functions
CRASH KINEMATICS For ngle A: opposite sine A = = hypotenuse djent osine A = = hypotenuse opposite tngent A = = djent For ngle B: opposite sine B = = hypotenuse djent osine B = = hypotenuse opposite tngent
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 informationSymmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedystte nlysis of high voltge trnsmission systems, we mke use of the perphse equivlent
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 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 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 informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in nonright ngled tringles. This unit
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 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 informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationfor all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx
Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion
More informationVECTORS VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors
1. Defnton A vetor s n entt tht m represent phsl quntt tht hs mgntude nd dreton s opposed to slr tht ls dreton.. Vetor Representton A vetor n e represented grphll n rrow. The length of the rrow s the mgntude
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationNaming the sides of a rightangled triangle
6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship
More informationProblems set # 3 Physics 169 February 24, 2015
Prof. Anhordoqui Problems set # 3 Physis 169 Februry 4, 015 1. A point hrge q is loted t the enter of uniform ring hving liner hrge density λ nd rdius, s shown in Fig. 1. Determine the totl eletri flux
More informationLecture 1  Introduction and Basic Facts about PDEs
* 18.15  Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1  Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationPythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides
Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of rightngled tringle is the longest side The hypotenuse is lwys opposite the rightngle 2 = 2 + 2 or 2 = 22 or 2 = 22 The re of the
More informationVECTORS. 2. Physical quantities are broadly divided in two categories viz (a) Vector Quantities & (b) Scalar quantities.
JMthemtics. INTRDUCTIN : Vectors constitute one of the severl Mthemticl systems which cn e usefully employed to provide mthemticl hndling for certin types of prolems in Geometry, Mechnics nd other rnches
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 informationReflection Property of a Hyperbola
Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the
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 informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More informationGeometry of the Circle  Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.
Geometry of the irle  hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion
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 informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 004 Exerise  0509 Exerise  093 Exerise  3 45 Exerise  4 6 Answer Key 78 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationForces on curved surfaces Buoyant force Stability of floating and submerged bodies
Stti Surfe ores Stti Surfe ores 8m wter hinge? 4 m ores on plne res ores on urved surfes Buont fore Stbilit of floting nd submerged bodies ores on Plne res Two tpes of problems Horizontl surfes (pressure
More informationDoes the electromotive force (always) represent work?
rxiv.org > physis > rxiv:1405.7474 Does the eletromotive fore (lwys) represent work?. J. Pphristou 1, A. N. Mgouls 1 Deprtment of Physil Sienes, Nvl Ademy of Greee, Pireus, Greee Emil: pphristou@snd.edu.gr
More informationThis enables us to also express rational numbers other than natural numbers, for example:
Overview Study Mteril Business Mthemtis 0506 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More informationSAMPLE EVALUATION ONLY
mesurement nd geometry topic 5 Geometry 5.1 Overview Why lern this? Geometry llows us to explore our world in very preise wy. uilders, rhitets, surveyors nd engineers use knowledge of geometry to ensure
More informationPythagoras theorem and surds
HPTER Mesurement nd Geometry Pythgors theorem nd surds In IEEM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of rightngled tringle. This result is known s Pythgors
More informationTopics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving rightangle triangles Sine and cosine rule
Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving rightngle tringles Sine nd osine rule Lelling rightngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite
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 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 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 information8.3 THE HYPERBOLA OBJECTIVES
8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an antiderivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous AntiDerivtive : An ntiderivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More informationThe Ellipse. is larger than the other.
The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationDEFINITION 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 informationCorrect answer: 0 m/s 2. Explanation: 8 N
Version 001 HW#3  orces rts (00223) 1 his printout should hve 15 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Angled orce on Block 01 001
More informationChapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
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 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 informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte singlevrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationPreLie algebras, rooted trees and related algebraic structures
PreLie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A prelie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers
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 informationMCH T 111 Handout Triangle Review Page 1 of 3
Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:
More informationBoard Answer Paper: October 2014
Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following suquestions: i. (D) ii..p
More informationFunctions. mjarrar Watch this lecture and download the slides
9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. OnetoOne Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides
More informationCBSEXII2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0
CBSEXII EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section
More informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 2942 9/11/04 Quntum Ciruit Model, SolovyKitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits  Universl Gte Sets A lssil iruit implements multioutput oolen funtion f : {0,1}
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 informationπ,π is the angle FROM a! TO b
Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two
More informationIdentifying and Classifying 2D Shapes
Ientifying n Clssifying D Shpes Wht is your sign? The shpe n olour of trffi signs let motorists know importnt informtion suh s: when to stop onstrution res. Some si shpes use in trffi signs re illustrte
More informationART LESSONS & EXERCISES
RT LESSONS & EXERISES Rmón Gllego. www.diujormon.wordpress.com www.diujormon.wordpress.com English  Pge 1 www.diujormon.wordpress.com English  Pge 2 STRTING UP WITH GEOMETRY. DEFINITION OF GEOMETRY Geometry
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 information16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings
Chpter 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings When, in the cse of tilted coordinte system, you brek up the
More informationGeometry AP Book 8, Part 2: Unit 3
Geometry ook 8, rt 2: Unit 3 IMRTNT NTE: For mny questions in this unit, there re multiple correct nswers, e.g. line segment cn e written s, RST is the sme s TSR, etc. Where pproprite, techers should e
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More information4. Statements Reasons
Chpter 9 Answers PrentieHll In. Alterntive Ativity 9. Chek students work.. Opposite sides re prllel. 3. Opposite sides re ongruent. 4. Opposite ngles re ongruent. 5. Digonls iset eh other. 6. Students
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationBasic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons.
si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)
More informationSection 11.2: The Law of Sines, from College Trigonometry: Corrected Edition by Carl Stitz, Ph.D. and Jeff Zeager, Ph.D. is available under a
Setion 11.: The Lw of Sines, from College Trigonometry: Correted Edition y Crl Stitz, Ph.D. nd Jeff Zeger, Ph.D. is ville under Cretive Commons AttriutionNonCommerilShreAlike 3.0 liense. 013, Crl Stitz.
More informationPhys 7221, Fall 2006: Homework # 6
Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 37 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which
More information( ) as a fraction. Determine location of the highest
AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
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 information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationr x a x b x r y a y b y r z a z b z. (310 to 312) s, multiply v by 1/s. (32) The Scalar Product The scalar (or dot) product of two vectors a (33)
REVIEW & SUMMARY 55 We net evlute ech term with Eq. 324, finding the direction with the righthnd rule. For the first term here, the ngle f etween the two vectors eing crossed is 0. For the other terms,
More informationLecture V. Introduction to Space Groups Charles H. Lake
Lecture V. Introduction to Spce Groups 2003. Chrles H. Lke Outline:. Introduction B. Trnsltionl symmetry C. Nomenclture nd symols used with spce groups D. The spce groups E. Derivtion nd discussion of
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 information20. Direct and Retrograde Motion
0. Direct nd Retrogrde Motion When the ecliptic longitude λ of n object increses with time, its pprent motion is sid to be direct. When λ decreses with time, its pprent motion is sid to be retrogrde. ince
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 informationa) Read over steps (1) (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.
Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You
More informationFor the flux through a surface: Ch.24 Gauss s Law In last chapter, to calculate electric filede at a give location: q For point charges: K i r 2 ˆr
Ch.24 Guss s Lw In lst hpter, to lulte eletri filed t give lotion: q For point hrges: K i e r 2 ˆr i dq For ontinuous hrge distributions: K e r 2 ˆr However, for mny situtions with symmetri hrge distribution,
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 informationare coplanar. ˆ ˆ ˆ and iˆ
SMLE QUESTION ER Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor. ii The question pper onsists of 6 questions divided into three Setions, B nd C. iii Question
More information6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR
6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationCIT 596 Theory of Computation 1. Graphs and Digraphs
CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege
More informationThermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R
/10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?
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 information