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
VECTOR ALGEBRA. Syllabus :
MV VECTOR ALGEBRA Syllus : Vetors nd Slrs, ddition of vetors, omponent of vetor, omponents of vetor in two dimensions nd three dimensionl spe, slr nd vetor produts, slr nd vetor triple produt. Einstein
More informationMarch 26, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
Mrh 26, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm D shown in the Figure Sine its digonls hve equl length,, the
More informationApril 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.
pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm
More informationOn 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 informationVectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a
Streth lesson: Vetors Streth ojetives efore you strt this hpter, mrk how onfident you feel out eh of the sttements elow: I n lulte using olumn vetors nd represent the sum nd differene of two vetors grphilly.
More information2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180.
SECTION 81 11 CHAPTER 8 Setion 8 1. There re n infinite numer of possile tringles, ll similr, with three given ngles whose sum is 180. 4. If two ngles α nd β of tringle re known, the third ngle n e found
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 information3. 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 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 informationTHREE DIMENSIONAL GEOMETRY
MD THREE DIMENSIONAL GEOMETRY CA CB C Coordintes of point in spe There re infinite numer of points in spe We wnt to identif eh nd ever point of spe with the help of three mutull perpendiulr oordintes es
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 informationGM1 Consolidation Worksheet
Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up
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 information, where P is called the tail and Q is called the nose of the vector.
6 VECTORS 1 INTRODUCTION TO VECTOR ALGEBRA 11 Slrs nd Vetors Slr: A slr is quntity tht hs only mgnitude ut no diretion Slr quntity is expressed s single numer, followed y pproprite unit, eg length, re,
More information3. Vectors. Vectors: quantities which indicate both magnitude and direction. Examples: displacemement, velocity, acceleration
Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 57 3. Vectors Vectors: quntities which indicte both mgnitude nd direction. Exmples: displcemement, velocity,
More informationChapter 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 ndimensionl
More informationPYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS
PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides
More informationPrecalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as
Dte: 6.1 Lw of Sines Syllus Ojetie: 3.5 Te student will sole pplition prolems inoling tringles (Lw of Sines). Deriing te Lw of Sines: Consider te two tringles. C C In te ute tringle, sin In te otuse tringle,
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 informationIMPOSSIBLE NAVIGATION
Sclrs versus Vectors IMPOSSIBLE NAVIGATION The need for mgnitude AND direction Sclr: A quntity tht hs mgnitude (numer with units) ut no direction. Vector: A quntity tht hs oth mgnitude (displcement) nd
More information3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.
3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationThings 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 informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationChapter 3. Vector Spaces. 3.1 Images and Image Arithmetic
Chpter 3 Vetor Spes In Chpter 2, we sw tht the set of imges possessed numer of onvenient properties. It turns out tht ny set tht possesses similr onvenient properties n e nlyzed in similr wy. In liner
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
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 informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 09765840 Volume 6, Numer (04), pp. 89 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3dimensional vectors:
Vectors 1232018 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 2dimensionl vectors: (2, 3), ( )
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
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 information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationTrigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis  The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
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 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 informationLESSON 11: TRIANGLE FORMULAE
. THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.
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 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 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 information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the solled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
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 informationProbability. b a b. a b 32.
Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PEhing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationInstructions to students: Use your Text Book and attempt these questions.
Instrutions to students: Use your Text Book nd ttempt these questions. Due Dte: 16092018 Unit 2 Chpter 8 Test Slrs nd vetors Totl mrks 50 Nme: Clss: Dte: Setion A Selet the est nswer for eh question.
More informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationIn rightangled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
Mth 3329Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then
More informationTrigonometry and Constructive Geometry
Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties
More information( ) { } [ ] { } [ ) { } ( ] { }
Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or
More informationMEP Practice Book ES19
19 Vectors M rctice ook S19 19.1 Vectors nd Sclrs 1. Which of the following re vectors nd which re sclrs? Speed ccelertion Mss Velocity (e) Weight (f) Time 2. Use the points in the grid elow to find the
More informationPROPERTIES OF TRIANGLES
PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties.
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 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 informationChapter 2. Vectors. 1 vv 2 c 2.
Chpter 2 Vetors CHAPTER 2 VECTORS In the first hpter on Einstein s speil theor of reltivit, we sw how muh we ould lern from the simple onept of uniform motion. Everthing in the speil theor n e derived
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More informationPart I: Study the theorem statement.
Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for
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 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 informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationComparing the Preimage and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) nglengle Similrity Theorem (.2) SideSideSide Similrity
More informationLearning Objectives of Module 2 (Algebra and Calculus) Notes:
67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under
More informationPolyphase Systems. Objectives 23.1 INTRODUCTION
Polyphse Systems 23 Ojetives eome fmilir with the opertion of threephse genertor nd the mgnitude nd phse reltionship etween the three phse voltges. e le to lulte the voltges nd urrents for threephse Yonneted
More informationInstructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.
ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s
More informationMathematics. 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 informationTable of Content. c 1 / 5
Tehnil Informtion  t nd t Temperture for Controlger 032018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................
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 informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationVector differentiation. Chapters 6, 7
Chpter 2 Vectors Courtesy NASA/JPLCltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higherdimensionl counterprts
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 informationSpacetime and the Quantum World Questions Fall 2010
Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36  out 3% () P (x 1
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for rightngled tringles
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 informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most wellknown nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationSimilar Right Triangles
Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems
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 informationat its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle.
Notes 6 ngle Mesure Definition of Rdin If circle of rdius is drwn with the vertex of n ngle Mesure: t its center, then the mesure of this ngle in rdins (revited rd) is the length of the rc tht sutends
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 informationUNIT 31 Angles and Symmetry: Data Sheets
UNIT 31 Angles nd Symmetry Dt Sheets Dt Sheets 31.1 Line nd Rottionl Symmetry 31.2 Angle Properties 31.3 Angles in Tringles 31.4 Angles nd Prllel Lines: Results 31.5 Angles nd Prllel Lines: Exmple 31.6
More informationA 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 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 informationMath Lesson 45 The Law of Cosines
Mth1060 Lesson 45 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude
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 informationHOMEWORK FOR CLASS XII ( )
HOMEWORK FOR CLASS XII 89 Show tht the reltion R on the set Z of ll integers defined R,, Z,, is, divisile,, is n equivlene reltion on Z Let f: R R e defined if f if Is f oneone nd onto if If f, g : R
More informationPolyphase Systems 22.1 INTRODUCTION
22 Polyphse Systems 22.1 INTRODUTION n genertor designed to develop single sinusoidl voltge for eh rottion of the shft (rotor) is referred to s singlephse genertor. If the numer of oils on the rotor is
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 informationTrigonometric Functions
Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds
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 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 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 informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
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 informationInspiration and formalism
Inspirtion n formlism Answers Skills hek P(, ) Q(, ) PQ + ( ) PQ A(, ) (, ) grient ( ) + Eerise A opposite sies of regulr hegon re equl n prllel A ED i FC n ED ii AD, DA, E, E n FC No, sies of pentgon
More information