Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:


 Reynold Randall
 2 years ago
 Views:
Transcription
1 Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos Fo emple, if point X lies long AB (ie position vectos of A nd B e known), position vectos of X cn e found if the tio of AX: XB is known. OA OB Fo AX: XB : s shown in the digm elow, OX using RT. A X B O Cse : If the point lies OUTSIDE two known position vectos This pplies to the scenio whee the point X lies on the line AB poduced s shown in the figue elow: A B X O 9 M Teo
2 Summ: Vectos 9 M Teo Using RT, students should get OX OA OB. Do note tht the suject (i.e. OB) is lws the point in the middle. The point X cn e found mking OX s the suject, i.e. OA OB OX ) (. Note: () The use of RT is ve common in A Levels question. () If, RT ecomes the midpoint theoem () The vectos used in the RT e lws POSITION VECTORS! ) Definition of scl nd vecto poduct The fomul (o definition) of Scl dot poduct is θ cos. The fomul (o definition) of Vecto coss poduct is θ sin. LHS of oth fomuls cn e detemined using vecto opetions. Fo emple, if nd, & ) ( Note: () Both vectos & must point in the sme diection. Refe to lectue notes to check wht I men! () will poduce vecto tht is mutull pependicul to nd. This is ve useful in ppliction polems involving plnes. ) Vecto line eqution A vecto line eqution cn e otined in two ws using: () Two known position vectos OR () A position vecto nd vecto (o diection vecto) pllel to the line.
3 Summ: Vectos 9 M Teo A vecto line eqution cn e epessed in two foms: () Pmetic fom i.e. l :, whee is n point on the line, is given point on the line, is diection vecto pllel to the line nd is the pmete. Note: the vecto line eqution esemles the stight line eqution i.e. m c fom A Mth. () Ctesin fom i.e. letting nd enging the pmetic fom mking the suject, the Ctesin eqution of the line cn e otined. Fo emple, If : l ) / ( ) / ( Hence, the Ctesin fom is Note: Students e dvised to know how to convet etween the two foms of the vecto line equtions. 4) Vecto plne eqution A vecto plne eqution cn e otined in thee ws using: () Thee known position vectos OR () Two known position vectos nd one vecto (o diection vecto) pllel to the plne OR () One known position vecto nd two vectos (o diection vectos) pllel to the plne A vecto line eqution cn e epessed in thee foms: () Pmetic fom i.e. c Π :, whee is n point on the plne, is given point on the plne, nd c e diection vectos pllel to the plne nd & e the pmetes. () Ctesin fom e.g. c d, whee,, c nd d e constnts. () Scl Dot Poduct fom i.e. n n, whee is n point on the plne, is given point on the plne, n is the vecto pependicul to the plne (i.e. lso clled the noml
4 Summ: Vectos vecto). This is the most useful nd esiest to use of the thee foms. Most pplictions equie the plne to e in this fom. Specil Note: It is ve useful to know how to chnge the foms of the vecto plne eqution, especill when solving pplictions polems involving vectos. Students cn convet the vecto plne equtions in the following mnne: Pmetic fom Scl dot poduct fom Ctesin fom Pmetic fom Scl dot poduct fom Ctesin fom The figue ove cn e eplined s follows: ) Pmetic fom cn e conveted to scl poduct fom, then to Ctesin fom, in this sequence. ) Scl poduct fom cn e conveted to Ctesin fom onl. c) Ctesin fom cn e conveted to scl poduct fom. d) Ctesin fom cn e conveted to pmetic fom. Note: It lso mens tht thee is no w to convet scl poduct fom to pmetic fom! Convesions () to (c) e eltivel es, which should e documented in most lectue notes. Hee, I will pesent the convesion of Ctesin fom diectl to pmetic fom. Fo emple, conside the Ctesin eqution of plne in the fom 4. Now, letting s the suject, eqution ecomes 4  (*) 4 9 M Teo
5 Summ: Vectos 9 M Teo Note: Students cn lso mke o s the suject. Now, we cn let & which will give us the two pmetes in the pmetic fom. Eqution (*) thus ecomes, 4 Since ( ) 4 Π : Hence, which is point on the plne. The two vectos pllel to the plne e &. ) Applictions of vectos ) Finding ngles The ke to finding ngles is to use the diection vectos of the line nd the plne (i.e. noml vecto). To detemine the ngles, students should use the scl dot poduct fomul i.e. θ cos θ cos Note: Usull cute ngles e equied, hence the fomul cn e modified to
6 Summ: Vectos cos θ Students should ecll tht if cos θ is positive, the ngle must lie in the fist qudnt (fom A Mth tigonomet) i.e. cute ngle. To find the ngle, simpl eplce vectos nd with the ppopite diectionl nd/o noml vectos. i) Finding ngle etween two lines Vectos nd e the diection vectos of ech line. ii) Finding ngle etween plne nd line Vectos nd cn e the noml vecto of the plne nd diection vecto of the line. The ngle etween them is thus 9  θ. See figue elow: Noml vecto of plne θ Line Angle equied iii) Finding ngle etween two plnes Vectos nd e the noml vectos of the plnes. ) Finding point of intesection i) Between two lines The point of intesection cn e found equting the two vecto line equtions nd solving fo the vlues of the two pmetes (e.g. nd ). Note: Thee equtions cn e fomed, ut onl two unknowns e solved. The thid eqution is to check fo consistenc of the vlues of nd found. If the thid eqution is not stisfied, this 6 9 M Teo
7 Summ: Vectos 7 9 M Teo will impl tht thee is no intesection etween the two lines. Lectue notes should contin this emple (plese efe). ii) Between line nd plne The point of intesect will lie on oth the line nd the plne. Since the point lie on the line, it must tke the vecto eqution of the line. Fo emple, if the intesect lie on : l, the position vecto of the point of intesect must e. To find the point of intesect, students e dvised to chnge the vecto plne eqution (if given in the ctesin o pmetic fom) to the scl dot poduct fom i.e. n n. Fo emple, to find the point of intesection etween the line nd plne elow, 4 & : l Hee, the position vecto of the intesection must e, s eplined elie. B fist chnging the vecto plne eqution to the scl dot poduct fom i.e. 4 4 Net, cn e eplced since the point will lso lie on the plne. Note: is n point on the plne Thus, 4, whee cn e found.
8 Summ: Vectos iii) Between two plnes When two plnes intesect, the will fom vecto line. To find the line, the plne equtions should e in Scl dot poduct fom & Pmetic fom Fo emple, if we hve Π : 7 & Π : 8 6 B chnging Π to nd sustituting this into Π, we will get Hee, we will get eltionship etween nd. B eithe mking o s the suject nd sustituting it ck into eqution of Π (i.e. pmetic fom), the line of intesection cn e found Note: This method educes the eqution of Π to single pmete, which now defines vecto line eqution. Altentivel, students cn chnge the plne equtions into the Ctesin fom nd use GC pplets to define the eqution of the intesection. iv) Between thee plnes When plnes intesect, thee e thee possile outcomes The fom point (i.e. position vecto) The fom line (i.e. vecto line eqution) Thee is no intesect Students cn detemine the intesection etween thee plnes using GC pplets. Note: This section is elted to sstem of line equtions. Unde tht chpte, thee is usull unique solution (i.e. the intesect to fom point), ut unde vectos, eithe one of the outcomes is possile. 8 9 M Teo
9 Summ: Vectos 9 9 M Teo c) Finding foot of pependicul When finding foot of pependicul, students should ecll tht if two vectos e pependicul, thei dot poduct is eo (i.e. ). i) Between point nd line To find the foot of pependicul, we fist must constuct diection vecto etween the point (i.e. position vecto) nd the foot of pependicul (i.e. ed point). Since the foot of pependicul lies on the line, it must tke the eqution of the line. Fo emple, if the line eqution is : l nd the given point (i.e. OA) is. Let N e the foot of pependicul ON The diection vecto is 4 OA ON AN Since AN is pependicul to the line AN, which llows ou to solve fo nd susequentl solve fo ON. Foot of pependicul Given point
10 Summ: Vectos Note: is the diection vecto of the line. Once the foot of pependicul is found, the shotest distnce (i.e. pependicul distnce) cn lso e found, tking the modulus of the diection vecto. ii) Between point nd plne Given point Noml vecto Foot of pependicul, ON To find the foot of pependicul (ON) to the plne, we cn fist constuct line contining the given point nd pllel to the noml vecto of the plne. Since the foot of pependicul is lso point on the line, it must tke the eqution of the line. The sme point (i.e. foot of pependicul) is lso point on the plne, hence we cn now sustitute the eqution into the eqution of the plne, which should e in the scl dot poduct fom. Fo emple, if eqution of plne is Π : nd the point is. 4 To find foot of pependicul, fist constuct the line contining nd pllel to 4 i.e. l : 4 9 M Teo
11 Summ: Vectos Since ON lies on the line, ON. Also, since ON lso lies on the plne, thus 4. The position vecto of the foot of pependicul cn e found once is 4 4 solved. Note: The vecto line is intentionll intoduced to detemine the foot of pependicul mens of intesecting line with plne. d) Finding length of pojection Most JC lectue notes contin the fomul to detemine the length of pojection of diection vecto onto nothe diection vecto. The fomul is Length of pojection ˆ Students e dvised to efe to lectue notes fo elevnt emples! Miscellneous These e othe points which m lso ppe in em, ut not s popul s those mentioned ove: i) Finding distnce of plne fom the oigin B modifing the scl dot poduct fom of the plne, the distnce etween the plne nd the oigin cn e found e.g. Π : n n n n Dividing oth sides n, o nˆ nˆ, whee nˆ is the unit vecto n n of noml vecto of the plne. The distnce etween the plne nd the oigin is thus tht the plne is elow the oigin. nˆ. If the distnce is negtive, it implies Note: This ppoch cn lso e used to detemine the pependicul distnce etween two plnes i.e. knowing the distnce of ech plne fom the oigin, the pependicul distnce is simpl the diffeence etween the two distnces. Cn ou visulie? 9 M Teo
12 ii) Finding e of tingle using vecto poduct iii) Finding eflection Summ: Vectos Ae of tingle ( )( )sinθ The midpoint theoem cn e used to find the position vecto of the eflected point. Refe to digm. Two position vectos must e povided in ode to solve. This method lso woks if the point is eflected out the plne. Point to e eflected Ais o point of eflection Reflected point O 9 M Teo
Intro to Circle Geometry By Raymond Cheong
Into to Cicle Geomety By Rymond Cheong Mny poblems involving cicles cn be solved by constucting ight tingles then using the Pythgoen Theoem. The min chllenge is identifying whee to constuct the ight tingle.
More informationr (1+cos(θ)) sin(θ) C θ 2 r cos θ 2
icles xmple 66: Rounding one ssume we hve cone of ngle θ, nd we ound it off with cuve of dius, how f wy fom the cone does the ound stt? nd wht is the chod length? (1+cos(θ)) sin(θ) θ 2 cos θ 2 xmple 67:
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationMAGNETIC FIELD AROUND CURRENTCARRYING WIRES. point in space due to the current in a small segment ds. a for field around long wire
MAGNETC FELD AROUND CURRENTCARRYNG WRES How will we tckle this? Pln: 1 st : Will look t contibution d to the totl mgnetic field t some point in spce due to the cuent in smll segment of wie iotsvt Lw
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationCurvature. (Com S 477/577 Notes) YanBin Jia. Oct 8, 2015
Cuvtue Com S 477/577 Notes YnBin Ji Oct 8, 205 We wnt to find mesue of how cuved cuve is. Since this cuvtue should depend only on the shpe of the cuve, it should not be chnged when the cuve is epmetized.
More information(Ch. 22.5) 2. What is the magnitude (in pc) of a point charge whose electric field 50 cm away has a magnitude of 2V/m?
Em I Solutions PHY049 Summe 0 (Ch..5). Two smll, positively chged sphees hve combined chge of 50 μc. If ech sphee is epelled fom the othe by n electosttic foce of N when the sphees e.0 m pt, wht is the
More informationOrbits and Kepler s Laws
Obits nd Keple s Lws This web pge intoduces some of the bsic ides of obitl dynmics. It stts by descibing the bsic foce due to gvity, then consides the ntue nd shpe of obits. The next section consides how
More informationr Curl is associated w/rotation X F
13.5 ul nd ivegence ul is ssocited w/ottion X F ivegence is F Tody we define two opetions tht cn e pefomed on vecto fields tht ply sic ole in the pplictions of vecto clculus to fluid flow, electicity,
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationCircles and Tangents with Geometry Expressions
icles nd Tngents with eomety xpessions IRLS N TNNTS WITH OMTRY XPRSSIONS... INTROUTION... 2 icle common tngents... 3 xmple : Loction of intesection of common tngents... 4 xmple 2: yclic Tpezium defined
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationTransformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) YanBin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationVectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationMoment and couple. In 3D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r
Moment and couple In 3D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita
More informationChapter 23 Electrical Potential
hpte Electicl Potentil onceptul Polems [SSM] A poton is moved to the left in unifom electic field tht points to the ight. Is the poton moving in the diection of incesing o decesing electic potentil? Is
More informationLaplace s Equation on a Disc
LECTURE 15 Lplce s Eqution on Disc Lst time we solved the Diichlet poblem fo Lplce s eqution on ectngul egion. Tody we ll look t the coesponding Diichlet poblem fo disc. Thus, we conside disc of dius 1
More informationExam in physics, Elgrunder (Electromagnetism), 20140326, kl 9.0015.00
Umeå Univesitet, Fysik 1 Vitly Bychkov Em in physics, Elgunde (Electomgnetism, 146, kl 9.15. Hjälpmedel: Students my use ny book(s. Mino notes in the books e lso llowed. Students my not use thei lectue
More informationScalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra
Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationMath 1105: Calculus II (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 5
Mth 5: Clculus II Mth/Sci mjos) MWF m / pm, Cmpion 35 Witten homewok 5 6.6, p. 458 3,33), 6.7, p. 467 8,3), 6.875), 7.58,6,6), 7.44,48) Fo pctice not to tun in): 6.6, p. 458,8,,3,4), 6.7, p. 467 4,6,8),
More informationN V V L. R a L I. Transformer Equation Notes
Tnsfome Eqution otes This file conts moe etile eivtion of the tnsfome equtions thn the notes o the expeiment 3 witeup. t will help you to unestn wht ssumptions wee neee while eivg the iel tnsfome equtions
More informationON THE CHINESE CHECKER SPHERE. Mine TURAN, Nihal DONDURMACI ÇİN DAMA KÜRESİ ÜZERİNE
DÜ Fen Bilimlei Enstitüsü Degisi Sı 9 Ağustos 9 On The Chinese Cheke Sphee M. Tun N. Donumı ON THE CHINESE CHECKER SHERE Mine TURAN Nihl DONDURMACI Deptment of Mthemtis Fult of Ats n Sienes Dumlupin Univesit
More informationCHAT PreCalculus Section 10.7. Polar Coordinates
CHAT PeCalculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationv o a y = = * Since H < 1m, the electron does not reach to the top plate.
. The uniom electic ield between two conducting chged pltes shown in the igue hs mgnitude o.40 N/C. The plte seption is m, nd we lunch n electon om the bottom plte diectl upwd with v o 6 m/s. Will the
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationVectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m
. Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationRandom Variables and Distribution Functions
Topic 7 Rndom Vibles nd Distibution Functions 7.1 Intoduction Fom the univese of possible infomtion, we sk question. To ddess this question, we might collect quntittive dt nd ognize it, fo emple, using
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationUnit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.
Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationAMPERE S LAW. by Kirby Morgan MISN0138
MISN0138 AMPERE S LAW by Kiby Mogn 1. Usefullness................................................ 1 AMPERE S LAW 2. The Lw................................................... 1. The Integl Reltionship...............................
More information32. The Tangency Problem of Apollonius.
. The Tngeny olem of Apollonius. Constut ll iles tngent to thee given iles. This eleted polem ws posed y Apollinius of eg (. 6070 BC), the getest mthemtiin of ntiquity fte Eulid nd Ahimedes. His mjo wok
More informationTrigonometry in the Cartesian Plane
Tigonomet in the Catesian Plane CHAT Algeba sec. 0. to 0.5 *Tigonomet comes fom the Geek wod meaning measuement of tiangles. It pimail dealt with angles and tiangles as it petained to navigation astonom
More informationG.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS
G.GMD.1 STUDENT NOTES WS #5 1 REGULAR POLYGONS Regul polygon e of inteet to u becue we begin looking t the volume of hexgonl pim o Tethedl nd to do thee type of clcultion we need to be ble to olve fit
More informationAdaptive Control of a Production and Maintenance System with Unknown Deterioration and Obsolescence Rates
Int J of Mthemtic Sciences nd Appictions, Vo, No 3, Septembe Copyight Mind Rede Pubictions wwwjounshubcom Adptive Conto of Poduction nd Mintennce System with Unknown Deteiotion nd Obsoescence Rtes Fwzy
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More information(1) continuity equation: 0. momentum equation: u v g (2) u x. 1 a
Comment on The effect of vible viscosity on mied convection het tnsfe long veticl moving sufce by M. Ali [Intentionl Jounl of Theml Sciences, 006, Vol. 45, pp. 6069] Asteios Pntoktos Associte Pofesso
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationScreentrade Car Insurance Policy Summary
Sceentde C Insunce Policy Summy This is summy of the policy nd does not contin the full tems nd conditions of the cove, which cn be found in the policy booklet nd schedule. It is impotnt tht you ed the
More informationNURBS Drawing Week 5, Lecture 10
CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu
More information69. The Shortest Distance Between Skew Lines
69. The Shortest Distnce Between Skew Lines Find the ngle nd distnce between two given skew lines. (Skew lines re nonprllel nonintersecting lines.) This importnt problem is usully encountered in one
More informationGRAVITATION 1. BASIC FORCES IN NATURE
GRAVITATION. BASIC ORCES IN NATURE POINTS TO REMEMBER. Bsing on the ntue nd eltive stength the bsic foces in ntue e clssified into fou ctegoies. They e ) Gvittionl foce ) Electomgnetic foce 3) Stong Nucle
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chpte The lectic Field II: Continuous Chge Distibutions Conceptul Poblems [SSM] Figue 7 shows n Lshped object tht hs sides which e equl in length. Positive chge is distibuted unifomly long the length
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationLAPLACE S EQUATION IN SPHERICAL COORDINATES. With Applications to Electrodynamics
LALACE S EQUATION IN SHERICAL COORDINATES With Appitions to Eetodynmis We hve seen tht Lpe s eqution is one of the most signifint equtions in physis. It is the soution to pobems in wide viety of fieds
More informationGauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationSHORT REVISION SOLUTIONS OF TRIANGLE
FREE Download Study Package fom website: wwwtekoclassescom SHORT REVISION SOLUTIONS OF TRINGLE I SINE FORMUL : In any tiangle BC, II COSINE FORMUL : (i) b + c a bc a b c sin sinb sin C o a² b² + c² bc
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More information2.016 Hydrodynamics Prof. A.H. Techet
.016 Hydodynmics Reding #5.016 Hydodynmics Po. A.H. Techet Fluid Foces on Bodies 1. Stedy Flow In ode to design oshoe stuctues, suce vessels nd undewte vehicles, n undestnding o the bsic luid oces cting
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
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 informationDETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.
Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of
More informationA couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance, d. F A F B (= F A
5 Moment of a Couple Ref: Hibbele 4.6, edfod & Fowle: Statics 4.4 couple is a pai of foces, equal in magnitude, oppositely diected, and displaced by pependicula distance, d. d (=  ) Since the foces ae
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationBrillouin Zones. Physics 3P41 Chris Wiebe
Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationFigure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!
1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationTrigonometric Identities & Formulas Tutorial Services Mission del Paso Campus
Tigonometic Identities & Fomulas Tutoial Sevices Mission del Paso Campus Recipocal Identities csc csc Ratio o Quotient Identities cos cot cos cos sec sec cos = cos cos = cot cot cot Pthagoean Identities
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationVariable Dry Run (for Python)
Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 2050 minutes
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationMultiplication and Division  Left to Right. Addition and Subtraction  Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis  Do ll grouped opertions first. E cuse Eponents  Second M D er Multipliction nd Division  Left to Right. A unt S hniqu Addition nd Sutrction
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationFormulas and Units. Transmission technical calculations Main Formulas. Size designations and units according to the SIunits.
Fomuls nd Units Tnsmission technicl clcultions Min Fomuls Size designtions nd units ccoding to the SIunits Line movement: s v = m/s t s = v t m s = t m v = m/s t P = F v W F = m N Rottion ω = π f d/s
More informationOn Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.
C.Candan EE3/53METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More information