Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young CMBRIDGE UNIVERSITY ENGINEERING DEPRTMENT PRT I (Frst Yer) 009-00 Pper 4 : Mthemtl Methods Leture ourse : Fst Mths Course, Letures 8 Leturer : Prof. J. B. Young Shedule : Weeks 4, Mhelms 009 Reommended ook : Jmes G. Modern Engneerng Mthemts. rd Edton. ddson Wesley. 00.
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young VECTOR LGEBR Vetor fundmentls. Defnton nd s rules of mnpulton. Emples of vetors. Representng vetors n omponent form Vetor multplton. The slr produt. The vetor produt. The slr trple produt.4 The vetor trple produt Vetor representton of lnes nd plnes. Strght lnes n D. Plnes 4 Vetors nd mtres 4. Mtr-vetor multplton 4. The trnspose of mtr 4. The determnnt of mtr 4.4 Slr nd vetor produts s mtr opertons 4.5 Smultneous equtons nd the nverse of mtr Referene : Jmes G., Modern Engneerng Mthemts, Chpters & 4 to 4.4.
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young VECTOR FUNDMENTLS. Defnton nd s rules of mnpulton In mehns, Newton s Seond Lw s wrtten F m, mplyng tht when ody of mss m s ted on y fore F the ody eperenes n elerton n the sme dreton s the fore. Ths notton s fne so long s we re delng wth prolem n one sptl dmenson. When fed wth rel-world D prolem, however, we requre three slr equtons reltng the omponents of the fore n the three rtesn dretons (F, F y, F z ) to the orrespondng omponents of the elerton (, y, z ) : F m, Fy my, Fz mz. Vetor methods hve een developed s shorthnd so tht the three slr equtons n e repled y sngle vetor equton reltng the vetor fore F to the vetor elerton : F m Of ourse, the ppltons of vetor lger re not restrted to mehns. For our purposes, we tke the defnton of vetor s follows : Vetors re qunttes possessng oth mgntude nd dreton whh oey the prllelogrm rule of ddton. The prllelogrm rule s equvlent to ddng the vetors nose to tl so n prte t s eser to work n terms of trngle rule of ddton :
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young Severl vetors n e dded geometrlly y suessve ppltons of the trngle rule : Vetors re sutrted y ddng the negtve of the relevnt vetor : ( ). Thus, f nd re the sme vetors s n the dgrms ove : Vetors, lke slrs, oey the followng lger rules : ( ) ( ) k ( ) k k (where k s ny slr onstnt) In these notes : Vetors wll e represented y uprght old type : The mgntude of vetor wll e represented y pr of vertl lnes : vetor of unt length wll e represented y ret ^ : â s unt vetor prllel to. Two or more vetors whh re perpendulr to eh other re sd to e orthogonl. Orthogonl vetors of unt length re sd to e orthonorml. The vetors,, k re understood to e unt vetors n the dretons of the, y, z es of rght-hnded rtesn oordnte system.
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young. Emples of vetors () Poston reltve to spefed orgn. z r y The poston vetor r, llustrted wth respet to rtesn oordnte system. () Lner dsplement. Dsplements n e dded usng the trngle lw : km E km N km S Resultnt dsplement () Lner veloty nd lner elerton. (v) ngulr veloty nd ngulr elerton. The ngulr veloty vetor ω of rottng ody s the vetor whose : mgntude equls the ody s rte of rotton (e.g., n rdns per seond), nd dreton s tht n whh rght-hnd srew wth the sme spn s the ody would move through sttonry ork. ω v ω θ r O r snθ Vew wth vetor ω omng out of the pge Note tht v ω r snθ
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young (v) Fore. Fores n e dded usng the trngle lw. (v) The moment of fore. The vetor moment M of vetor fore F ppled t pont r out n orgn O s the vetor whose : mgntude equls the moment out O, nd dreton s tht n whh rght-hnd srew would move f twsted y the moment. Vetor moment M hs mgntude θ M F r sn θ O r F nd dreton nto the pge. (v) The dfferentl re on surfe. The dfferentl vetor re d s vetor whose : mgntude equls the re d of dfferentl element on the surfe, nd dreton s norml to, nd outwrd from, the sde of the surfe defned s postve. d d Qunttes tht hve mgntude nd dreton ut do not oey the prllelogrm lw re not vetors. For emple, the fnte ngulr rotton of ody s not vetor. Ths s euse the pplton of fnte rotton θ out the s followed y fnte rotton θ out the z s, sy, gves dfferent result from rotton θ out the z s followed y rotton θ out the s. However, the resultnt of two nfntesml rottons dθ nd dθ s ndependent of the order of pplton so tht ngulr velotes (of mgntude dθ /dt nd dθ /dt) do dd vetorlly (nd so ngulr veloty s vetor). It s mportnt to pprete tht the mgntude nd dreton of vetor s ndependent of the oordnte system used to desre t. Ths s the gret dvntge of workng wth vetors. For emple, the sttement s ompletely ndependent of ny oordnte system. Ths mens tht we n mnpulte vetors usng the rules of vetor lger wthout spefyng oordnte system. It s only when we ome to the stge of nsertng numerl vlues tht s t neessry to hoose oordnte system. 4
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young. Representng vetors n omponent form We n epress the mgntude nd dreton of vetor numerlly wth respet to ny hosen o-ordnte system. For rtesn system, we set up mutully perpendulr es, y nd z, nd defne, nd k to e vetors of unt length long the es. vetor n then e epressed s lner omnton of the vetors, nd k : k z y, nd re lled the omponents of. These lone re suffent to spefy the vetor one the oordnte es hve een defned, so s often represented y, or (,, ). The onventon s to use rght-hnded set of es, otned s follows. Drw es nd ( nd y) t rght ngles n plne. Drw s (z) norml to the plne n the dreton rght-hnded srew would move were s rotted to le on top of s. It wll now e found tht s s n the dreton rght-hnded srew would move f s were rotted to le on top of s. Smlrly, s s n the dreton rght-hnded srew would move f s were rotted to le on top of s. Note the yl order : rotted to gves, rotted to gves, rotted to gves. ddton of vetors s omplshed y ddng omponents. Thus, ) ( ) ( ) k ( or, lterntvely, or, more omptly, (,, ) 5
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young VECTOR MULTIPLICTION. The slr produt The slr produt of two vetors nd s wrtten where θ s the nluded ngle. os θ nd s defned y, θ The slr produt n e thought of s : The mgntude of multpled y the omponent of n the dreton of, or The mgntude of multpled y the omponent of n the dreton of. In formng the slr produt, we tke two vetors nd end up wth slr ( sngle numer). It n e seen from the defnton tht the order of the vetors does not mtter, lso, the slr produt oeys the so-lled dstrutve lw,. ( ) If nd re orthogonl, os θ 0 so 0. If nd re prllel, os θ so. If nd re nt-prllel, os θ so. The unt vetors n rtesn oordntes therefore oey the followng reltonshps : k k k k 0 6
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young 7 If we epress the vetors nd n rtesn form nd multply out, we otn, z z y y z y z y ) ( ) ( k k In prtulr, ) ( ) ( z y z y z y k k It s very esy to form the slr produt numerlly s shown y ths emple : ) ( ) ( 4) ( 4 The slr produt s n nvrnt. Ths mens t tkes the sme vlue whtever oordnte system s used. If, n the ove emple, we hd deded to use D polr rther thn rtesn oordntes, the omponents of nd would e dfferent ut the slr produt would stll e. s physl emple of the use of the slr produt suppose vetor fore F moves ody vetor dstne d, not neessrly n the sme dreton s the fore : The omponent of F prllel to the dreton of moton does work on the ody ut the norml omponent mkes no ontruton. Hene, the work done y the fore s, W F os θ d F d θ F d
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young. The vetor produt The vetor produt of two vetors nd (wrtten ross ) s defned to e vetor whose mgntude s gven y sn θ or nd pronouned where θ s the nluded ngle etween the vetors (suh tht θ < 80 when the tls of the vetors re mde to onde). The dreton of s perpendulr to oth nd, suh tht f we rotte to le on top of, the dreton of moton of rght-hnded srew s n the dreton of. θ In formng the vetor produt we tke two vetors nd otn nother vetor. It n e seen from the defnton tht the order of the vetors s mportnt nd tht, Lke the slr produt, the vetor produt oeys the dstrutve lw, ( ) If nd re orthogonl, sn θ so. If nd re prllel or nt-prllel, sn θ 0 so 0. The unt vetors n rtesn oordntes oey the followng reltonshps : k, k, k k, k, k 0, 0, k k 0 8
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young 9 Epressng the vetors nd n rtesn omponents nd multplyng out, we otn, ) ) k ( k ( k - ( - ( - ( ) ) ) Lke the slr produt, the vetor produt s nvrnt; ts vlue does not depend on the oordnte system used. In the ove form, the rtesn epnson for s qute dffult to rememer. The result n e found n the mthemts dt ook ut the esest wy of gettng the rtesn omponents orret s to rememer the representton of s determnnt (see lter f you don t know wht determnnt s) : k k k - ( - ( - ( ) ) ) The followng re emples of the use of the vetor produt : () The vetor re of prllelogrm : The mgntude of the re of the prllelogrm s sn θ. The vetor re (dreton out of the pge) s, (The order s mportnt) θ vetor re
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young () The vetor moment of fore : Moment out O s requred d r θ F Pont of pplton of fore O The mgntude of the moment out O of the fore F, ppled t pont on the ody defned y the poston vetor r, s M F r sn θ. The vetor moment (dreton nto the pge) s, M r F (The order s mportnt) () The veloty t pont n rottng ody : ω v O θ r ω r snθ When ody rottes wth ngulr veloty ω s shown, the mgntude of the veloty t the pont spefed y the poston vetor r s v ω r sn θ. The vetor veloty (dreton s shown) s, v ω r (The order s mportnt) 0
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young. The slr trple produt φ The prllelepped shown ove s defned y the three vetors, nd. The re of the se s (wth dreton upwrds ). The volume V of the prllelepped ( postve slr quntty) s therefore, V os φ ( ) ( ) s known s the slr trple produt. In ft, the rkets re unneessry s the epresson ould not e evluted n ny other wy. The volume n e evluted usng ny two of the vetors to form the se. Hene, we dedue tht hngng the vetors round n yl order does not hnge the vlue of the slr trple produt euse, ( ) ( ) ( ) V Chngng the yl order, however, hnges the sgn of the slr trple produt, From the ove reltonshps we otn, ( ) ( ) ( ) V ( ) ( ) ( ) whh shows tht we n nterhnge the nd the n slr trple produt wthout ffetng the vlue. Beuse of ths, the slr trple produt s sometmes wrtten [,,] s there n e no mguty. Note tht [,,] 0 f the three vetors re oplnr (the prllelepped volume s zero).
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young.4 The vetor trple produt The other possle omnton of three vetors s the vetor trple produt, ( ). The result of ths operton gves vetor whh s orthogonl to the vetors nd. We n derve n mportnt vetor dentty for smplfyng lultons where the vetor trple produt ppers. We do ths y onsderng the (or ) omponent n rtesn epnson : [ ( ) ] ( ) ( ) ( - ) ( - ) ( ) ( ) ( ) ( ) [( ) ( )] Note the lever trk n lne 4 where ws oth dded nd sutrted. The sme result s otned for the nd omponents (y symmetry), nd hene we otn the dentty : ( ) ( ) ( ) If, nsted, the frst two vetors re rketed, we esly fnd tht, ( ) ( ) ( ) ( ) s n emple of vetor mnpulton, we smplfy the epresson ( ) ( ) : ( ) ( ) ( ) (defnng ) ( ) (nterhngng the nd the ) [ ( )] [( ) ( )] (usng the vetor trple produt dentty) ( ) ( ) ( ) ( ) tully, n prte, t s rther eser to wrte, ( ) ( ) sn θ ( os θ) (!)
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young VECTOR REPRESENTTION OF LINES ND PLNES. Strght lnes n D z ( ) r y Two seprte ponts defne strght lne n D spe. Let nd e the poston vetors of the two known ponts nd let r e the poston vetor of generl pont on the lne. The vetor equton of the lne s then, r λ( ) where λ s slr prmeter. When λ 0, r, nd when λ, r. By hoosng vlues of λ etween nd, every pont on the lne s otned. If t (for tngentl) s unt vetor n the dreton of the lne, then the ove equton n e wrtten, r µ t where µ s dfferent slr prmeter (unless hppens to equl t). Note tht lnes n lwys e represented n terms of ust one slr prmeter. The vetor equton for lne s relly three slr equtons rolled nto one. To fnd the slr rtesn form, we deompose the vetor equton y wrtng, λ( ) y λ( ) z λ( ) more ompt wy of presentng these equtons s, y z ( )
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young z r y nother wy of wrtng the vetor equton of lne s, r where, s shown n the dgrm, s vetor tngentl to the lne nd s prtulr vetor norml to the lne (whh then defnes the lne). To onvne yourself tht the ove equton relly does represent lne, note tht the poston vetor of pont on the lne must stsfy, Equtng the two epressons for nd usng the dstrutve property of the vetor produt n reverse, we fnd, (r ) 0 Hene, (r ) s vetor prllel to nd so we n wrte r λ whh s the equton of lne, n the dreton of, pssng through the pont defned y the poston vetor. 4
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young. Plnes ( ) µ ( ) ( ) r λ ( ) O Three ponts defne plne so long s they do not le on sngle strght lne. If, nd re the poston vetors of the three known ponts, then t n e seen from the dgrm tht the poston vetor r of generl pont on the plne s gven y, r λ( ) µ( ) where λ nd µ re slr prmeters tht n tke ny vlues etween nd. Note tht plnes n lwys e spefed n terms of two slr prmeters. n (r ) d O r n lterntve form of the vetor equton for plne n e otned n terms of vetor n norml to the plne. s n e seen from the dgrm the equton s, (r ) n 0 Epndng n slr form, we otn the well-known rtesn equton for plne, n n y n z n If n s unt norml vetor ( n n n ), t n e seen from the dgrm tht the slr produt n d represents the perpendulr dstne d of the orgn O from the plne. 5
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young Two plnes defne lne unless they re prllel : Suppose we hve two plnes defned y the equtons r p nd r q. By omprson wth the generl equton, we see tht nd re vetors norml to the plnes (not unt vetors, neessrly). The lne of nterseton wll therefore e prllel to the vetor nd n e wrtten, r λ ( ) where s the poston vetor of ny pont lyng on the lne nd λ s slr prmeter. Note tht f the vetor produt 0 the two normls re n the sme dreton so the plnes re prllel nd do not nterset. Furthermore, f, p q the plnes re one nd the sme; nsted of lne, we hve plne of nterseton. Three plnes defne pont unless ther normls re oplnr : The pont of nterseton of three plnes n e found y solvng the three smultneous equtons, r p, r q nd r s. In rtesn omponent form, these re : y z p y z q y z s 6
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young Geometrlly, we epet unque soluton ( sngle pont of nterseton) unless : () The plnes re ll prllel ut seprte, so there s no nterseton, or () The plnes re ll the sme, so there s plne of nterseton, or () The three plnes nterset long sngle lne, so there s lne of nterseton, or (v) Two plnes re prllel, so there s no ommon nterseton, see elow, or (v) We get Tolerone stuton, see elow, so there s no ommon nterseton. In ll these spel ses, the normls to the three plnes n e moved so tht they re oplnr. Therefore, the volume of the prllelepped they form s zero. Hene, we n reognse the spel ses euse 0. 7
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young 4 VECTORS ND MTRICES 4. Mtr-vetor multplton The most generl wy n whh the omponents of vetor n e trnsformed y lner opertons nto the omponents of vetor s s follows : where, et re onstnt oeffents. In settng up shorthnd notton to epress these reltonshps, we defne the mtr s the rry of oeffents : n lso e wrtten where (,, ) enumertes the rows nd (,, ) the olumns. n esy wy of rememerng how the susrpts hnge s provded y the mnemon : Wrtng the vetors nd n olumn formt (.e., s mtres), we defne mtrvetor multplton so tht, In order to stsfy these equltes, the th omponent of the olumn vetor must e otned y multplyng the th row of the mtr y the olumn vetor n the sense tht, The mtr s sd to operte on the vetor to gve the vetor. Wth the understndng tht mtr-vetor multplton s mpled, we n now wrte, 8
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young 4. The trnspose of mtr The trnspose t of mtr s defned to e mtr n whh rows nd olumns re nterhnged (row of eomes olumn of t nd so on). Thus, t or t slr n e thought of s mtr, so t trnsposes nto tself, λ t λ. vetor s usully thought of s mtr, lthough ths s only onventon. If we need the omponents set out s row, we smply wrte the vetor s t, t ( ) Ths s the sme vetor ut epressed n dfferent wy. 4. The determnnt of mtr Every squre mtr hs numer ssoted wth t lled determnnt, wrtten or sometmes det. One gn, ths s done s onvenent shorthnd notton. The determnnt of mtr s defned s, The determnnt of mtr s then gven y, The determnnts re otned y rossng out the frst row nd eh olumn n turn of the orgnl mtr. They re lled the mnors of the determnnt. Wth the relevnt sgn n front (,, ) they re lled the oftors. The representton ove s lled epnson out the frst row. In ft, determnnt n e epnded out ny row or olumn so long s proper ttenton s gven to the sgns. 9
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young When the determnnts re multpled out, we fnd, ( ) ( ) ( ) ( ) ( ) Note the rght-hnded yl order n the frst rket nd the left-hnded yl order n the seond rket. n esy wy of otnng ths epresson s to wrte out the mtr wth the frst nd seond olumns repeted : d d d d4 d5 d6 * * * * * * * * * * * * * * * The vlue of the determnnt s then the sum of the three produts of the three numers down the dgonls, d, d, d, mnus the sum of the three produts of the three numers down the dgonls d4, d5, d6. 4.4 Slr nd vetor produts s mtr opertons In order to epress the slr produt s the operton of mtr on vetor, we trnspose the frst vetor nto mtr. Thus, Note tht, t ( ) t t Epressng the vetor produt s the operton of mtr on vetor s rther more omplted. Thus, 0 0 0 0
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young Note tht the determnnt of ths mtr s zero so the mtr hs no nverse. Hene, gven the vetor nd the vetor produt we nnot solve for. One onsequene of ths s tht f, we nnot ssume tht. smlr stuton ests for slr produts. If nd re equl, we nnot ssume tht. s mentoned erler, the vetor produt n lso e epressed s determnnt : k k Ths leds us on to prove tht the slr trple produt n lso e epressed s determnnt : ( ) k k ( k) k It hs lredy een shown tht s the volume of the prllelepped formed y the vetors, nd, nd tht the sgn s hnged f we swp nd. From ths we dedue the rule for determnnts tht f two rows (or two olumns) re swpped, the sgn of the determnnt hnges. On the other hnd, yl permutton of rows nd olumns leves the vlue of determnnt unhnged. 4.5 Smultneous equtons nd the nverse of mtr So fr, the mtr hs een presented s wy of representng three lner equtons reltng eh omponent of vetor to the omponents of nother vetor. More generlly, however, ny set of three lner equtons n e wrtten n mtr notton. For emple, the equtons desrng the nterseton of three plnes ( r p, r q nd r s) re, y z p y z q y z s
Engneerng Prt I 009-0, Pper 4, Mthemtl Methods, Fst Course, J.B.Young In mtr form these re wrtten, z y s q p Formlly, we n otn the poston vetor of the pont of nterseton ( y z) t y fndng the nverse of the mtr nd usng t to operte on the vetor (p q s) t. Thus, z y s q p Ths proedure ssumes tht the mtr relly does hve n nverse. However, f the normls to the three plnes re oplnr, the volume of the prllelepped formed s zero, 0. But lso equls the determnnt of the mtr nd so we dedue tht f the determnnt of mtr s zero the nverse does not est. It s mportnt to note tht the equton nvolvng the nverse represents the forml soluton to the orgnl mtr equton. In prte, one never tully solves set of smultneous equtons y lultng the nverse of the mtr, prtulrly on omputer. Rther one uses muh more effent numerl tehnque suh s Gussn elmnton. s n emple, we hek to see f the followng equtons hve unque soluton : 9 4 6 The equtons n e wrtten, 6 4 0 9 Usng the dgonl multplton trk for lultng the determnnt of the mtr, we fnd, 6 4 0 9 [ 0 6 ( ) 4 9 ] [9 0 4 ( ) 6] 0 So the nverse of the mtr does not est nd there s no unque soluton to the equtons.