LINEAR ALGEBRA, VECTOR ALGEBRA AND ANALYTICAL GEOMETRY

Transcription

1 ФЕДЕРАЛЬНОЕ АГЕНТСТВО ПО ОБРАЗОВАНИЮ Государственное образовательное учреждение высшего профессионального образования «ТОМСКИЙ ПОЛИТЕХНИЧЕСКИЙ УНИВЕРСИТЕТ» VV Koev INER GEBR VECTOR GEBR ND NYTIC GEOMETRY TetBoo Рекомендовано в качестве учебного пособия Редакционно-издательским советом Томского политехнического университета Издательство Томского политехнического университета 9

2 UDС 7 VV Koev e lge Veto lge d ltl Geomet Tetoo Toms: TPU Pess 9 pp Ths tetoo ossts of pts devoted to the mthemtl methods of e lge d ltl Geomet sed o the veto lss tehque The s oepts e epled emples d llustted fgues The tetoo s helpful fo studets who wt to udestd d e le to use mt opetos solve sstems of le equtos le eltve postos of fgues tsfom oodte sstems d so o The tetoo s desged to Eglsh speg studets Revewed : V Kl Pofesso of the Hghe Mthemts Deptmet TPU DS Koev VV -9 Toms Polteh Uvest -9

3 PREFCE Ths tetoo s teded fo studets who hve led studed s mthemts d eed to stud the methods of hghe mthemts It oves thee otet es: e lge Veto lge d ltl Geomet Eh pt ots s mthemtl oeptos d epls ew mthemtl tems M useful emples d eeses e peseted the tetoo epled d llustted emples d eeses The e lge tops lude mt opetos detemts d sstems of le equtos I the seto Veto lge m tteto s pd to the geometl ppltos of veto opetos The veto ppoh s osdeed to e s fo dsusso of lss polems of ltl Geomet The utho welomes ede s suggestos fo mpovemet of futue edtos of ths tetoo 6

4 CONTENTS Pefe Cotets INER GEBR Chpte MTRICES Bs Deftos 7 Mt Opetos 8 Tpes of Mtes Koee Delt Smol Popetes of Mt Opetos 6 Chpte DETERMINNTS Pemuttos d Tspostos Detemt Geel Defto Popetes of Detemts Detemt Clulto Chpte INVERSE MTRICES Thee emms 6 Theoem of Ivese Mt 8 Emples 9 Clulto of Ivese Mtes Elemet Tsfomtos Chpte SYSTEMS OF INER EQUTIONS Mt R Bs Coepts Guss Elmto 6 Emples 7 Homogeeous Sstems of e Equtos Emples Cme s Rule 6 Cme s Geel Rule 7

5 VECTOR GEBR Chpte VECTORS Bs Deftos 6 Geometl Itepetto 6 Vetos Thee-Dmesol Spe 6 e Veto Opetos 6 Poeto of Veto Gve Deto 6 Popetes of e Veto Opetos 6 Resoluto of Vetos to Compoets 6 Retgul Othogol Bss 6 e Depedee of Vetos 66 Veto Bses 68 Sl Podut of Vetos 69 Popetes of the Sl Podut 7 Some Emples 7 Deto Coses 7 Veto Podut 7 Popetes of the Veto Podut 7 Some Emples 7 6 The Sl Tple Podut 7 6 Popetes of the Sl Tple Podut 76 6 Some Emples 77 7 Tsfomto of Coodtes Ude Rotto of the Coodte Sstem 79 7 Rotto of the Ple oud the -s 8 NYTIC GEOMETRY Chpte 6 STRIGHT INES 6 Equtos of les 8 6 es Ple 8 6 gle Betwee Two es 86 6 Dste Fom Pot to e 89 6 Reltve Posto of es 9

6 Chpte 7 PNES 7 Geel Equto of Ple 9 7 Equto of Ple Pssg Though Thee Pots 9 7 Othe Foms of Equtos of Ple 9 7 gle Betwee Two Ples 9 7 Dste Betwee Pot d Ple Reltve Posto of Ples Reltve Posto of Ple d e gle Betwee Ple d e 98 Chpte 8 Qudt Cuves 8 Cles 99 8 Ellpses 8 Popetes of Ellpses 8 Hpeols 8 Popetes of Hpeols 6 8 Pols 9 8 Summ Refeees 6

7 INER GEBR Mtes 7 Mtes Mtes llow us to opete wth s osstg of m umes futos o mthemtl sttemets ust s f we opete wth sevel tems Mtes hve wde pplto dffeet hes of owledge fo ste mthemts phss ompute see d so o Mtes llow us to solve sstems of od equtos o sets of dffeetl equtos to pedt the vlues of phsl quttes qutum theo to ept messges the Iteet d so o I ths hpte we dsuss the s oepts of the mt theo todue mt htests d stud some mt ppltos The mpott popostos e poved d llustted emples Bs Deftos mt s etgul of umes lge smols o mthemtl futos povded tht suh s e dded d multpled odg to et ules Mtes e deoted uppe se lettes: B C The se of mt s gve the ume of ows d the ume of olums mt wth m ows d olums s lled m mt (pooue m-- mt) The umes m d e the dmesos of the mt Two mtes hve the sme se f the dmesos e equl Emples: mt mt mt 7 s os B C 8 os s Memes of mt e lled ts mt elemets o etes The et the -th ow d the -th olum of mt s deoted o The suspts dte the ow fst d the olum seod I the emples ove the oldfe elemets e d mt wth oe ow s lled ow mt: ( ) K

8 Mtes mt wth oe olum s lled olum mt: m I the geel fom mt s wtte s follows: m m shot fom of ths eqult s m m sque mt hs s m ows s olums the ume of whh detemes the ode of the mt tht s mt s the mt of the -th ode Mt Opetos Eqult of Mtes Two mtes d B e equl f the hve the sme ses d the elemets e equl ps tht s B fo eh p of dees { } Sl Multplto mt m e multpled o the ght o left sl qutt λ The podut s the mt B λ (of the sme se s ) suh tht fo eh { } λ To multpl mt sl multpl eve mt elemet tht sl Emple: et The 8

9 Mtes The Sum of Mtes If d B e mtes of the sme se the the sum B s the mt C suh tht fo eh p { } To dd mtes dd the oespodg mt elemets Emple: et 7 6 d B The B Multplto of Row Colum et e ow mt hvg s m elemets s olum mt B I ode to multpl B t s eess to multpl the oespodg elemets of the mtes d to dd up the poduts Smolll B ( ) K Thus multplg ow mt olum mt we ot ume te we wll show tht ume e osdeed s mt K To multpl two-ow mt the K olum mt B ( B) M we multpl eh ow of the olum of B I ths se the podut B s the followg mt: B ( B ) B B K K Smll the multplto of m-ow mt -olum mt geetes the m mt 9

10 Mtes Mt Multplto The podut of two mtes d B s defed f d ol f the ume of elemets ow of equls the ume of oes olum of B et e m l mt d B e l mt The the podut B s the m mt suh tht ts et the -th ow d the -th olum s equl to the podut of the -th ow of d the -th olum of B If we deote the ows of K d the olums of B B B K the C B m To fd the elemet mt ( B B B ) B B m B B B m B B B m B the -th ow d the -th olum of the C B multpl the -th ow of the -th olum of B: B Note : The smol otto mes the podut of two equl sque mtes: Smll K Note : I geel the podut of mtes s ot ommuttve: l B B Emples: ) Fo eh of the followg mtes B C D ( ) d F deteme whethe t equls the mt o ot Soluto: The dmesos of oth mtes C d D dffe fom oes of Theefoe C d D Thee e two mtes B d F whh osst of the sme elemets s d hve the sme ode Howeve the oespodg etes of d B e ot equl ps d so B The mt F stsfes ll odtos of mt eqult tht s F

11 Mtes ) et d B Solve fo X the mt equto X B Soluto: 6 8 B X ) Gve two mtes ( ) d fd the mt poduts B d B B Soluto: ( ) ) ( B ( ) 8 B ) et d B Fd the dffeee etwee mt poduts B B d Soluto: 9 ) ( ) ( B ) ( ) ( B B B

12 Mtes ) Fd Soluto: f d so o Tpes of Mtes I sque mt the elemets wth e lled the dgol mt elemets The set of the etes foms the ledg (o pple) dgol of the mt sque mt s lled dgol mt f off-dgol elemets e equl to eo o smolll M fo ll : M Idett mtes I e sque mtes suh tht I d I Compe these mt equltes wth the oespodg popet of el umes: d Theoem: dett mt I s dgol mt whose dgol elemets e equl to ut: I M M Ths theoem s poved the followg seto

13 Mtes Emples: ) It s ot dffult to vef tht Theefoe d d d d d s the dett mt of the seod ode ) et e mt The d mt s lled eo-mt (-mt) f t ossts of ol eo elemets: fo eh { } I shot fom eo-mt s wtte s : M O M B the defto of eo-mt d tht s eo-mt hs ust the sme popetes s the ume eo Howeve f the podut of two mtes s equl to eo t does ot me tht t lest oe of the mtes s eo-mt Fo ste oth mtes d B e o-eo mtes whle the podut s eo-mt: B

14 Mtes sque mt hs tgul fom f ll ts elemets ove o elow the ledg dgol e eos: ll fo > o fo < Emples: Uppe-tgul mt owe-tgul mt B 7 Gve m mt the tspose of s the m mt T oted fom tehgg ts ows d olums Ths mes tht the ows of the mt e the olums of the mt d vse ves: T ( ) Fo ste the tspose of sque mt 7 T s 7 T ; s lled smmet mt f s equl to the T tspose of : The emples elow llustte the stutue of smmet mtes: d e T T R R d S d f S e f sque mt s lled sew-smmet mt f s equl to the opposte of ts tspose: The emple elow shows the stutue of sew-smmet mt: T

15 Koee Delt Smol The Koee delt smol s defed the fomul f δ f Mtes The delt smol els summto ove oe of the dees suh epessos s δ δ δ δ d so o Fo ste the sum δ m ot ol oe oeo tem δ whle ll the othe tems e equl to eo euse of δ fo If If the δ < o > the δ ewse f the δ Othewse f Emples: < o > the δ δ K 9 δ howeve δ Now we esl pove the ove-metoed theoem of dett mt: I δ M O M

16 Mtes The theoem sttes tht the I δ s dett mt Theefoe we hve to pove tht I fo mt Poof: et e t m mt d δ e the sque mt 6 of the -th ode The the mt podut I s the mt of the sme se s B the defto of the mt podut d vew of the popetes of the delt smol we ot tht I ) δ ( fo eh p of dees { } The eqult of the oespodg mt elemets mples the eqult of the mtes: I Popetes of Mt Opetos Popetes volvg ddto Fo mt thee ests the opposte mt ( ) suh tht ( ) If d B e mtes of the sme se the B B If B d C e mtes of the sme se the ( B) C (B C) The tspose of the mt sum s the sum of the tspose of the mtes: T T T ( B) B The ove popetes of mtes esult fom the popetes of el umes The poofs e left to the ede Popetes volvg Multplto et e mt If λ d µ e sl quttes the λ ( µ ) ( λ µ ) et d B e two mtes suh tht the podut B s defed If λ s sl qutt the λ ( B) ( λ ) B ( λ B) et B d C e thee mtes suh tht ll eess multpltos e ppopte The (B)C (BC)

17 7 Mtes et d B e two mtes suh tht the podut B s defed The T T T ( B ) B If d B e two dgol mtes of the sme ode the B B Popetes ) d ) smpl esult fom the popetes of el umes d the defto of the sl multplto To pove Popet we hve to show tht the oespodg elemets of the two mtes ( B) C d (BC) e equl B the defto the mt elemet the -th ow d the -th olum of the mt B s ( B ) ll l The mt elemet the -th ow d the -th olum of the mt ( B)C e epessed s (( B ) C) ( B) C B hgg the ode of summto we ot (( B) C) l l l l l ( BC) l l ( ( BC)) The eqult of the oespodg mt elemets s stsfed tht mples the eqult of the mtes: ( B ) C ( BC) To demostte Popet we tsfom the et the -th ow d the T -th olum of the mt (B) I vew of the defto of the tspose of mt T ( B) ( B) T T T T B T B T l l T l l l ( B Thus ( B) d ( B ) oe the odtos of eqult of mtes Popet s sed o the followg esos: ) dgol mtes e smmet oes; ) the podut of dgol mtes s dgol mt Theefoe we eed ol to show tht ( B ) ( B) Ideed ( B ) ( B) T T )

18 Mtes Popetes volvg ddto d Multplto et B d C e thee mtes suh tht the oespodg poduts d sums e defed The ( B C) B C ( B) C C BC et d B e two mtes of the sme se If λ s sl the λ ( B) λ λ B To pove Popet osde the elemet o the -th ow d the -th olum of the mt ( B C) B the defto of the mt podut d vew of the ddto popetes we hve ( ( B C)) ( B C) fo eh p of dees { } ( ( B) ) ( C) Theefoe the mtes (B C) d (B C) e equl ( B C) The eqult of the mtes ( B)C d (C BC) e pove sml w: (( B) C) ( B) C ( ( C) ) ( BC) ( C BC) The oespodg mt elemets e equl ps Hee the mtes e equl Popet esults fom the popetes of el umes The poof e pefomed the ede Shot Summ: Opetos wth mtes suh s ddto d multplto hve sml popetes s tht wth usul el umes Numel mtes of the fst ode e tepeted s usul el umes tht s The set of mtes s geelto of the set of el umes 8

19 Emples: ) et ( ) B d C B stghtfowd poedue show tht (B)C (BC) Soluto: B ( ) ( 7) ( B) C ( 7) ( 9 7) B C 8 ( B C) 9 7 ( B) 8 ( ) ( ) C Mtes ) et d B e two mtes of the seod ode Vef the dett T T T ( B ) B Soluto: Fd the mt podut of d B d the tspose of B: B ( B) T T T The fd the mt podut B to see tht ( B ) B : B T T ) et ( ) f d Fd f () Soluto: The mt-logue of the ume s the dett mt I Theefoe f ( ) I T T T 9

20 Detemts DETERMINNTS Pemuttos d Tspostos pemutto of elemets of set of odeed elemets s oe-to-oe tsfomto of the set oto tself et S e the odeed set of the tul umes fom to : S { K } pemutto of S s the set of the sme umes ged ptul w: K } { K } { pemutto s lled tsposto f the ode of two elemets of the set s hged ut ll othe elemets em fed Emple of pemutto: { } { } Emple of tsposto: { } { } Eve pemutto of odeed elemets e epessed though sequee of sevel tspostos Fo ste pemutto { } e peseted the sequee of the followg tspostos: { } { } { } { } It s sd tht pemutto of S ots the veso of elemets f < d > The totl ume of vesos detemes the veso pt of the pemutto tht tes o two vlues: ethe eve o odd pemutto s lled eve pemutto f t ots eve ume of vesos Ths mes tht eve pemutto s fomed eve ume of tspostos of S odd pemutto ots odd ume of vesos Ths mes tht odd pemutto s sequee of odd ume of tspostos of S d Emple: The pemutto { } of { } ots thee vesos: d d d s odd se t

21 Detemts Theoem tsposto hges the veso pt of pemutto Poof: It s ot dffult to see tht the tsposto of eghog elemets d hges the veso pt of gve pemutto The tsposto of elemets d e epessed s the sequee of ( ) tspostos Rell tspostos of the elemet wth the eghog elemet o the ght of { } : we get the pemutto The - tspostos of the elemet elemet o the left of { } : wth the eghog we get the desed pemutto The totl ume ( ) of the tspostos s odd ume d hee the veso pt of the pemutto s hged Theoem Gve the set S { K } thee e! dffeet pemuttos of S Poof: Cosde t pemutto of S The fst posto e dspled of elemets The seod posto e dspled of the emg elemets The thd posto e dspled of the emg elemets d so o The -th posto e dspled the est sgle elemet Theefoe thee e ( )( )! ws to get ew pemutto of the elemets of S

22 Detemts Emple: The set S { } ossts of thee elemets d so the ume of dffeet pemuttos s! 6: { } { } { } { } { } { } ) The pemuttos { } { } d { } e eve se eh of them s sequee of eve ume of tspostos of the elemets of S: { } { } { } { } { } { } I tems of vesos the pemuttos { } { } d { } e eve se eh of them ots eve ume of vesos of the elemets Fo ste the pemutto { } ots two vesos of the elemets: d se s o the left fom d > d se s o the left fom d > ) ewse the pemuttos { } { } d { } e odd se eh of them s sequee of odd ume of tspostos of the elemets of S I ptul the pemutto { } s the tsposto of the elemets d of S I tems of vesos the pemutto { } s odd se t ots the odd ume of the vesos: d se s o the left fom d > d se s o the left fom d > d se s o the left fom d > The pemutto { } ots the veso of the elemets d The pemutto { } ots the veso of the elemets d

23 et Geel Defto e sque mt of the ode d let the odeed set of the fst tul umes Cosde the followg podut of mt elemets: P{ K ( ) } Detemts S { } e K () whee } s pemutto of S d P } s the { veso pt of the pemutto } Tht s ( ) fo { eve pemutto d ( ) P fo odd oe: P{ K } ) sg{ K } ( { P Epesso () s the podut of mt elemets suh tht eh ow d eh olum of s peseted oe d ol oe ts elemet odg to Theoem thee e! dffeet pemuttos of S eh of whh geetes the podut of tpe () The sum of poduts () ove ll possle pemuttos { } s lled the detemt of the mt : det { K } P{ K ( ) () It s deoted the etwee vetl s: K det () M M Sum () ots! tems () wth eve d odd pemuttos fft-fft The detemt s ve mpott htest of the mt s ule t s mpott ol whethe the detemt of gve mt equls eo o ot Fo ste the vese mt of ests f d ol f det Do ot ofuse the detemt of mt wth the mt tself! Whle umel mt s of umes ume ut ot of umes } det s some sgle

24 Detemts Ptul ses mt of the fst ode ots ol oe elemet The detemt of tht mt s equl to the mt elemet tself: det et e sque mt of the seod ode: Thee est the followg two pemuttos of { } : { } d { } The pemutto { } s eve se t does ot ot vesos whle the pemutto {} s odd se two elemets fom the veso These pemuttos geete two poduts of the elemets wth opposte sgs d the sum of whh gves the detemt of : If mt hs the thd ode the we hve to osde ll possle pemuttos of the set { } Thee est the followg s pemuttos of { } : { } { } { } { } { } { } The pemuttos { } { } d { } e eve se eh of them ots eve ume of vesos of elemets The pemuttos { } { } d { } e odd se thee e odd umes of vesos of elemets these pemuttos (See detls the ove emple) Theefoe To ememe ths fomul ppl the Sus Rule whh s show the fgue elow

25 Detemts The elemets o dgol o t the vetes of tgul fom the podut of thee elemets If the se of the tgle s pllel to the ledg dgol of the mt the podut eeps the sg; othewse the podut hges the sg Popetes of Detemts The detemt of the tspose of s equl to the detemt of the gve mt : det T det Poof: Ths popet esults fom the detemt defto se oth detemts osst of the sme tems Multplg ow o olum of detemt ume λ multples the detemt tht ume: Ths mes tht the ommo fto of ow (olum) e te out Poof: Eve tem of the sum det { K } K ( ) P{ K ots oe d ol oe elemet of ow d olum of the mt Theefoe f the ow (o olum) s multpled ume eh tem s multpled tht ommo fto }

26 Detemts 6 The detemt hges the sg f two ows (o olums) of mt e tehged: Poof: B Theoem tsposto hges the veso pt of gve pemutto Theefoe eh tem of the sum } { } { ) ( det P K K K hges ts sg If mt hs eo-ow o eo-olum the the detemt s equl to eo: Poof: Eve podut of the sum P det ) ( } { } { K K K ots eo fto d so equls eo If mt hs two equl ows (o olums) the the detemt s equl to eo:

27 Detemts 7 Poof: et two detl ows (o olums) e tehged The Popet the detemt hges the sg O the othe hd the ows (o olums) e equl d hee the detemt eeps ts vlue: det det det 6 If two ows (o olums) of mt e popotol to eh othe the the detemt s equl to eo: Poof: Multplg the -th ow of the mt the ostt of popotolt we ot the detemt wth equl ows 7 If eh elemet of ow (olum) of detemt s the sum of two etes the Poof: } { } { } { } { ) ( ) ( P K K K K K K K K K K

28 Detemts 8 detemt holds ts vlue f ow (olum) s multpled ume d the s dded to othe oe: Poof: The detemt o the ght hd e epessed s the sum of two detemts oe of whh ots two popotol ows Theefoe the detemt equls eo 9 et d B e sque mtes of the sme ode The the detemt of the podut s equl to the podut of the detemts: det( B) det det B The detemt of tgul mt s equl to the podut of the elemets o the pple dgol: K I ptul the detemt of dett mt I equls the ut Poof: Fst thee s ol whh s o-eo elemet the fst olum Theefoe sum () ossts of eo tems fo ll vlues of eept fo Net we hve to goe the fst ow d hoose o-eo elemet o the seod olum Ol the elemet stsfes these odtos d so we set sum () ewse o the thd olum we te ol the elemet to get o-eo podut of elemets d so o Theefoe ll ppopte pemuttos of dees gve eo poduts of elemets eept fo the podut of the elemets o the pple dgol 8

29 Detemts Emples: s os ) et Fd det os s Soluto: s os det s os os s ) et Vef tht d Soluto: T det det det d d det T d d d ) Evlute Soluto: ) et d B Vef tht det B det det B Soluto: 7 det det B det det B 7 B det B ( ) 9

30 Detemts ) Evlute Soluto: Note tht The det f det (det ) det det 6) et Clulte: () det () (e) det( I) det () ) det( (d) det( ) Soluto: () The detemt of mt the tgul fom equls the podut of the pple dgol elemets Theefoe det ( ) () The detemt of the podut of mtes s equl to the podut of the detemts d so det (det ) ( ) 8 () et I e the dett mt of the thd ode The (d) ewse det( ) det(i ) det ( ) 6 det( ) det( I ) det ( ) ( ) (e) Smplf the mt ( I ) : I The detemt of ths mt equls eo: det( I )

31 Clulto of Detemts Detemts Methods of detemt lulto e sed o the popetes of detemts Hee we osde two methods whh eg omed togethe esult the most effet omputg tehque Epdg detemt ow o olum Befoe fomultg the theoem let us todue few deftos et e sque mt of the ode B emovg the -th ow d -th olum we ot sumt of hvg the ode ( ) The detemt of tht sumt s lled the mo of the elemet whh M s deoted ( ) It s deoted the smol : ( ) M The ofto of the elemet s defed s the mo wth the sg M The followg theoem gves sstemt poedue of detemt lulto The detemt of mt equls the sum of the poduts of elemets of ow of d the oespodg oftos: det K The ove theoem s ow s the epso of the detemt odg to ts -th ow Poof: B the defto det s the lge sum of the poduts K te wth the sgs

32 Detemts } { ) ( } { sg P K K ove ll possle pemuttos tht s } { K } { } { ) ( det P K K K Eh podut ots the elemet o the -th ow d -th olum Theefoe egoupg the tems the ove sum e epessed s the le omto of the elemets ( K ): det K Hee } { sg } { K B the theoem of veso pt of pemutto } sg{ ) ( } { sg Thee e ) ( vesos of the pemutto d so } { } sg{ ) ( } { sg } sg{ ) ( } { sg Howeve M } { } sg{ K s the mo of the elemet Theefoe s the ofto of the elemet M ) ( Se oth mtes d the tspose of hve equl detemts the theoem e fomulted tems of epdg detemt olum: The detemt of mt equls the sum of the poduts of elemets of olum of d the oespodg oftos: det K Due to the theoem gve detemt of the ode s edued to detemts of the ode ( )

33 Detemts Emples: ) Epd the detemt of the mt of the ode () the fst ow; () the seod olum Compe the esults Soluto: det det ( ( ) ( ) Both esults e detll equl ) Clulte the detemt ( ) ) ( 7 odg to the fst ow d the seod olum Soluto: The epso the fst ow elds 7 7 ( ) ( (7 ) ) ) ts epso Now epd the detemt odg to the seod olum: 7 ( ) 7 ( ) ( 9) 7( ) 7

34 Detemts Evluto of detemts elemet opetos o mtes B mes of elemet ow d olum opetos mt e edued to the tgul fom the detemt of whh s equl to the podut of the dgol elemets et us defe the elemet opetos I vew of the popetes of detemts tehques whh e developed fo ows m e lso ppled to olums I ode to lulte detemt oe m: Itehge two ows s esult the detemt hges ts sg Multpl ow oeo ume s osequee of ths opeto the detemt s multpled tht ume dd ow multpled ume to othe ow B ths opeto the detemt holds ts vlue We lso use the elemet opetos to get some ow o olum osstg of eo elemets eept fo oe elemet d the to epd the detemt tht ow (o olum) Emples: ) et B elemet ow d olum opetos o the mt edue the mt to the tgul fom d lulte det Soluto: det The detemt of the mt the tgul fom s equl to the podut of the elemets o the pple dgol Theefoe det

35 ) Evlute the detemt of the mt 6 7 Detemts Soluto: Fst tsfom the fst ow v elemet olum opetos Keepg the fst d lst olums sutt the fst olum multpled fom the seod oe d dd the fst olum multpled to the thd oe: det 7 6 The epd the detemt the fst ow: det 7 Tsfom the thd olum ddg the thd ow to the fst oe d suttg the thd ow multpled fom the seod ow: 9 Epd the detemt the thd olum: det 9 We stll te out the ommo fto fom the lst ow: 9 det (( 9) ( ) )

36 Ivese Mtes 6 Ivese Mtes et e sque mt mt s lled vese mt of f I whee I s dett mt If the detemt of mt s equl to eo the the mt s lled sgul; othewse f det the mt s lled egul If eh elemet of sque mt s epled ts ofto the the tspose of the mt oted s lled the dot mt of : T d Thee emms emm : Gve sque mt of the ode the sum of the poduts of the elemets of ow (o olum) d the oftos of othe ow (olum) s equl to eo: ) ( () d ) ( () Poof: To pove () osde ul mt ~ tht s oted fom the mt eplg the -th ow wth the -th oe: ~ Epd ~ det the -th ow: ~ ~ ~ det

37 Ivese Mtes The ol dffeee etwee mtes ~ d s the -th ow Howeve ~ the oftos do ot deped o the elemets o the -th ow d so ~ whh mples ~ det O the othe hd the mt ~ hs two equl ows Theefoe the popetes of detemts ~ det ( ) Sttemet () e pove sml w emm : The mt poduts d d d e dgol mtes tht s ( d ) ( ) Poof: If d the emm ( d ) ( ) ( d ) T T ( d ) emm : The dgol elemets of the mtes e equl to the detemt of the mt : ( d ) (d ) det d d d Poof: B the theoem of epso of detemts odg to ow det T ( d ) ewse the theoem of epso detemts olum elds det Hee the lemm T (d ) 7

38 Ivese Mtes Theoem of Ivese Mt Poof: Fo egul mt thee ests the uque vese mt: d det sgul mt hs o vese mt ssume tht thee ests vese of mt The I det det d hee det Theefoe sgul mtes hve o vese mtes ssume tht eh of the mtes d B s vese of : I d B B I The B B I B ( B ) I Theefoe thee ests the uque vese of Fd the vese of mt B the emm ( d ) f B the emm ( d ) det Comg the ove equltes we ot ( d ) δ det whee the delt smol δ deotes the mt elemets of dett mt Theefoe d I det ewse ( d ) I d hee det d det 8

39 Ivese Mtes 9 Emples of Clultos of Ivese Mtes Emple : Gve the mt fd the vese of Soluto: Fst lulte the detemt: 6 det Net fd the oftos of ll elemets: ) ( ) ( ) ( ) ( The fd the dot mt of : d T T Fll ot det Vefto: I d I Emple : et Fd the vese of Soluto: Clulte the detemt: 6 det Theefoe the gve mt s sgul d so t hs o the vese of

40 Ivese Mtes Emple : et Fd the vese of mt Soluto: ) To lulte the detemt of dd the fst ow douled to the seod ow The epd the detemt the seod olum det ( ) ( ) ) Fd the oftos of the elemets of the mt ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) Wte dow the dot mt of d ) The vese of mt s 6 T 6

41 Ivese Mtes ) Vefto: I ewse I Emple : et d B Solve fo X the mt equto X B Soluto: Note tht det tht s s egul mt Theefoe thee ests the vese of : B X Fd the vese of mt d T d det Thus 7 7 X Vefto: B X 7 7

42 Ivese Mtes Clulto of Ivese Mtes Elemet Tsfomtos et e egul mt The vese of e foud mes of the elemet tsfomtos of the followg eteded mt ) ( I whee I s the dett mt of the oespodg ode B mg use of elemet ow opetos we hve to tsfom the eteded mt to the fom ( I B ) The B The followg elemet tsfomtos (volvg ol ows) e ppled: ) Multplg ow oeo ume ) ddg ow to othe ow Emple: et Fd the vese of Soluto: Cosde the eteded mt ) ( I Multpl the seod ow the ume d the sutt the esult fom the fst ow: Sutt the fst ow fom the seod oe: Dvde the seod ow : The desed fom s oted d hee

43 Sstems of e Equtos Mt R m mt s sd to e the mt of f thee ests t lest oe egul sumt of ode ; eve sumt of hghe ode s sgul odg to the defto m{ m } Sstems of e Equtos The of mt e evluted pplg ust those elemet ow d olum opetos whh e used to smplf detemts tht s Itehgg two ows o olums Multplg ow (olum) oeo ume Multplg ow (olum) ume d ddg the esult to othe ow (olum) If ow o olum ossts of eos the t e omtted These opetos e sd to e elemet tsfomtos of mt Theoem: If mt ~ s oted fom elemet tsfomtos ~ the Poof: Itehgg two ows o two olums of mt hges the sg of the detemt Multplg ow (olum) oeo ume multples the detemt tht ume ddg ow (olum) to othe oe holds the mgtude of the detemt Theefoe ll sgul sumtes e tsfomed to sgul sumtes d egul sumtes e tsfomed to egul sumtes Hee the theoem B elemet tsfomtos of mt we t to ot s m eos s possle to edue the mt to the ehelo fom: Fo ste 7 6 s the mt of the edued ow ehelo fom The ume of the ows gves the of :

44 Sstems of e Equtos Emples: ) et 8 7 Fd the of Soluto: Sutt the fst d fouth ows fom the thd oe: dd the thd ow multpled sutle umes to the othe ows: 8 8 Suttg the fst ow fom the fouth ow d the ddg the seod ow to the fouth oe we ot futhe tsfomtos e ot eess euse the detemt of the ode s equl to eo ut thee s sumt of the thd ode the detemt of whh s oeo: ) ( ) ( Hee

45 M Deftos Cosde sstem of m le equtos wth uows: m m K K K m m Sstems of e Equtos Hee e umel oeffets; e ostts ( m) d e uows ( ) soluto of sstem () s set of vlues of the uows () tht edues ll equtos () to dettes If thee ests soluto of smulteous equtos the the sstem s lled osstet; othewse the sstem s osstet Usg mt multpltos sstem of equtos () e epeseted sgle mt equto X B whee s the oeffet mt osstg of ; the olum mt B s lled the o-homogeeous tem; X s the olum mt whose elemets e the uows : X B m m m m If the o-homogeeous tem B s equl to eo the the le sstem s lled the homogeeous sstem: X Two le sstems e lled equvlet f the hve the sme soluto set Elemet tsfomtos of the le sstem s the poess of otg equvlet le sstem fom the gve sstem the followg opetos: ) Itehge of two equtos ) Multplto of equto oeo ume ) ddto of equto multpled ostt to othe equto Eh of the ove opetos geetes equvlet le sstem

46 Sstems of e Equtos Two le sstems of equtos e equvlet f oe of them e oted fom othe the elemet tsfomtos pplg the le tsfomtos we t to fd equvlet sstem whh e ese solved Guss Elmto Cosde the ugmeted mt of sstem (): ( B) m m 6 m m Thee s oe-to-oe oespodee etwee the elemet tsfomtos of the le sstem d le ow opetos o the ugmeted mt Ideed: Itehgg two equtos of the sstem oespods to tehgg the ows of the ugmeted mt Multplto of equto oeo ume oespods to multplto of the ow tht ume ddto of two equtos of the sstem oespods to ddto of the ows of the mt The m de s the followg Fst tsfom the ugmeted mt to the uppe tgle fom o ow ehelo fom: ~ ~ ~ ~ ~ ~ ( B) M O M M ~ ~ M The wte dow the le sstem oespodg to the ugmeted mt the tgle fom o edued ow ehelo fom Ths sstem s equvlet to the gve sstem ut t hs smple fom Fll solve the sstem oted the method of susttuto If t s eess ssg pmet vlues to some uows Ths sstemt poedue of solvg sstems of le equtos elemet ow opetos s ow s Guss elmto

47 Sstems of e Equtos 7 Some Emples ) Solve the sstem elow Guss elmto: Soluto: Redue the ugmeted mt to tgle fom: The lte mt oespods to the sstem whh s equvlet to the tl sstem Now the soluto e esl foud: Thus we ot the soluto of the gve sstem X It s ot dffult to vef the vlues of the uows stsf ll the gve equtos: ) ( ) ( ) (

48 Sstems of e Equtos 8 ) Fd ll solutos of the sstem of equtos v Guss elmto Soluto: The sstem e epeseted the ugmeted mt ppl the le ow opetos: 7 9 The thd ow oespods to the equto 7 whh evdetl hs o solutos Theefoe the gve sstem s osstet ) Use Guss elmto to solve the sstem of equtos Soluto: B elemet tsfomtos the ugmeted mt e edued to the ow ehelo fom: 9 The edued mt hs the d oespods to the followg sstem of le equtos:

49 Sstems of e Equtos 9 9 The vle s osdeed to e t pmete egdless of the vlue of whh the emg vlues of edue ll equtos of the gve sstem to dettes d Fom the lst equto we fd 9 The we ot The geel soluto of the sstem X 9 9 depeds o the t pmete ptul vlue of gves ptul soluto of the sstem ssgg fo ste the eo vlues to the pmete we ot ptul soluto X Settg we ot othe ptul soluto 9 8 X Coluso: The gve sstem hs fte ume of solutos Soluto he: et us vef tht the set of vlues

50 Sstems of e Equtos 9 9 stsfes the gve sstem of equtos: Tht s tue Homogeeous Sstems of e Equtos homogeeous sstem of le equtos hs the followg fom; X () whee s the oeffet mt d X s the olum mt of the uows Evdetl homogeeous sstem hs the ptul soluto X M whh s lled the tvl soluto Theoem: If X d X e solutos of homogeeous sstem the le omtos of the solutos X X s lso soluto of the sstem Poof: B the odtos of the theoem X d X Fo ostts d X ( X ) X ( X )

51 Sstems of e Equtos ddg togethe the ove dettes we ot ) ( ) ( X X whh mples ( ) X X Hee the theoem Emples ) Use Guss elmto to solve the followg homogeeous sstem of equtos Soluto: B elemet tsfomtos the oeffet mt e edued to the ow ehelo fom The of ths mt equls d so the sstem wth fou uows hs fte ume of solutos depedg o oe fee vle If we hoose s the fee vle d set the the ledg uows d e epessed though the pmete The ove mt oespods to the followg homogeeous sstem 9 9 The lst equto mples Usg the method of susttuto we ot

52 Sstems of e Equtos Theefoe the geel soluto of the sstem s X To ot ptul soluto we hve to ssg some umel vlue to the pmete If we set the X 6 X Soluto he: The set of vlues of the uows edues equtos of the gve le sstem to the dettes: 6 ) et Fd the soluto of the homogeeous sstem of le equtos X Soluto: Tsfom the oeffet mt to the ow ehelo fom:

53 Sstems of e Equtos Se we hve to hoose two uows s the ledg uows d to ssg pmet vlues to the emg uows Settg d we ot the followg le sstem: Theefoe d ) ( Thus the gve sstem hs the followg geel soluto: X I vew of the mt popetes the geel soluto e lso epessed s the le omto of ptul solutos: X The ptul solutos X d X fom the sstem of solutos whh s lled the fudmetl set of solutos Thus X X X

54 Sstems of e Equtos ) et ) Solve the followg homogeeous sstem of le equtos X ) Epl wh thee e o solutos fte ume of solutos o etl oe soluto Soluto: Note tht homogeeous sstem s osstet d hs t lest the tvl soluto Tsfom the oeffet mt to the tgul o ow ehelo fom The of equls Theefoe thee e o fee vles d the sstem hs the tvl soluto ol Cme s Rule Thee s ptul se whe the soluto of sstem of le equtos e wtte the eplt fom The oespodg theoem s ow s Cme s Rule whose mpote s detemed ts ppltos theoetl vestgtos Cme s Rule: et X B () e sstem of le equtos wth uows If the oeffet mt s egul the the sstem s osstet d hs uque soluto set whh s epeseted the fomul: } { K D D () whee D det ; s the detemt of the mt oted eplg the -th olum of wth the olum mt B: D

55 D Sstems of e Equtos M M Poof: We hve to pove the followg sttemets: ) soluto s uque; ) fomuls () follow fom sstem (); ) fomuls () eld sstem () Se det thee ests the vese of Theefoe mt eqult () mples X B () B the theoem of vese mt fo egul mt thee ests uque vese mt d det tht poves the uqueess of soluto () The -th ow of d s fomed the oftos K of the elemets the -th olum of the mt The eqult () mples ( B) D M D ( ) The sum o the ght sde s the epso of the detemt tems of the elemets the -th olum Hee we hve oted the desed fomul: D D Now pove tht the set sstem () K } wth D { D D mples Multpl oth sdes of ths eqult d the sum the esult ove : D Itehge the ode of summto the epesso o the ght sde

56 Sstems of e Equtos 6 D (6) I vew of the theoem of vese mt det δ whee δ s the Koee delt The Koee delt tes w the summto ove epesso (6): D D D δ Hee we hve the desed le sstem of equtos: ( ) The theoem s pove Emple: Use Cme s Rule to solve the followg sstem of le equtos Soluto: D D D

57 9 9 D Theefoe Sstems of e Equtos D 86 D D D D D Compe ths soluto wth tht oted Guss elmto Emple p 6 Cme s Geel Rule Cme's Geel Rule fomultes the estee odto of soluto fo gve sstem of le equtos Cme's Geel Rule: sstem of m le equtos wth uows m m K K K m s osstet f d ol f the of the ugmeted mt s equl to the of the oeffet mt Poof: et e the oeffet mt d let e the ugmeted mt of the gve sstem We hve to pove tht () If the sstem s osstet the () If the the sstem s osstet To pove sttemet () we hve to ssume tht the sstem s osstet Cosde the ugmeted mt : M m M M If we sutt the fst olum multpled the seod olum multpled d so o fom the lst olum the we ot the mt of the sme s ( the theoem of mt ): m m m 7

58 Sstems of e Equtos M m M m 8 ( m ( m K M K m ) ) Se the sstem s osstet eh elemets of the lst olum equls eo Theefoe M m () Now suppose tht M m M M m M m It mes thee ests osgul sumt ~ of the mt whh we selet ledg uows d ssg pmet vlues to the emg ( ) fee uows The edued sstem of le equtos s equvlet to the tl sstem d Cme s Rule t hs uque soluto fo eh set of vlues of the fee uows Hee the theoem Cooll: ) If d equls the ume of uows the the soluto of the sstem s uque ) If < the thee est fte ume of solutos of the gve sstem Sttemet ) follows fom the Cme s Rule If < the the gve sstem s equvlet to the sstem of le equtos wth ledg uows fte ume of the vlues of the emg ( - ) uows leds to fte ume of solutos Emples: The sstem of le equtos s gve elow Fomulte the odtos o d mg the sstem to e osstet Soluto: Cosde the ugmeted mt d tsfom t to the edued ow ehelo fom

59 Sstems of e Equtos The sstem s osstet f Othewse oe of uows s pmet vle d the sstem hs fte ume of solutos Gve the edued ow ehelo fom of the ugmeted mt 7 fd the ume of solutos of the oespodg sstem It s ot eess to solve the sstem Soluto: The of the oeffet mt equls the of the ugmeted mt d equls the ume of the uows Hee the Cooll to Cme s Geel Rule the soluto s uque et sstem of le equtos e gve the ugmeted mt How m solutos hs the sstem? Soluto: If the whle B Cme s Geel Rule the sstem s osstet d so t hs o solutos If the whle the ume of the uows s So oe of the uows hs to e osdeed s pmete d the sstem hs soluto fo eh vlue of tht pmete Hee the sstem hs fte ume of solutos

60 Vetos VECTOR GEBR Vetos Bs Deftos thee-dmesol veto some oodte sstem s odeed tplet of umes tht oes et ules of ddto d multplto d tht e tsfomed ude otto of oodte sstem ust s the oodtes of pot The memes of the tplet e lled the oodtes of the veto ewse oe defe -dmesol veto Usull vetos e deoted oldfe lettes: The otto { } mes tht the umes d e the oodtes of the veto thee-dmesol oodte sstem Two vetos { } d { } e equl f the oodtes e espetvel equl tht s Note tht veto eqult s equvlet to the sstem of thee sl equltes fo the oodtes of the veto e veto opetos lude the multplto of veto sl qutt d the ddto of vetos If veto { } s multpled sl λ the λ s the veto suh tht λ λ λ The sum of two vetos } d } s the veto { { { } The dffeee etwee two vetos s defed tems of ddto: ( ) Theefoe 6

61 Geomet Itepetto Vetos Thee-Dmesol Spe Vetos Cosde etgul oodte sstem et { } e gve veto d P d P e two pots wth the oodtes ( ) d ( ) espetvel The pots P ( ) d P ( ) e seleted so s to stsf odtos Theefoe veto e tepeted s the deted le segmet P fom to P : P The oodtes of P P e equl to the dffeees etwee the oespodg oodtes of the pots P ) d P ) : ( P P P P P { } ( The pot P s the se of d s the hed The se of veto s lso lled the veto tl o the og of the veto The legth of veto P P s defed s the legth of the le segmet og P d P Note tht veto s qutt possessg oth mgtude d deto t oe The oldfe lette epesets veto qutt whle s the mgtude of the veto tht s s sl qutt etel defed umel vlue If veto os the og of the oodte sstem wth pot P( ) the t s lled the dus-veto of the pot P d deoted s 6

62 Vetos e Veto Opetos Eqult of Vetos B pllel tslto equl vetos should ode wth eh othe: Sl Multplto The legth of the veto λ s λ If λ > the s veto of the sme deto s : If λ < the veto λ hs the opposte deto wth espet to : The opposte veto of B s the veto B B The legth of ut veto equls ut If s o-eo veto the u s the ut veto the deto of The Sum of Two Vetos 6

63 The Dffeee Betwee Two Vetos Vetos I ode to sutt veto fom dd the opposte of to : ( ) Thus the dffeee etwee two vetos d s the veto suh tht Poeto of Veto Gve Deto et θ e gle etwee two vetos d The qutt Po osθ () s lled the poeto of o If θ s ute gle the the poeto s postve If θ s otuse gle the the poeto s egtve Oe esl pove tht If the deto s detemed the -s the the poeto of oto the -s equls the dffeee etwee the oodtes of the edpots of the veto: 6

64 Vetos Popetes of e Veto Opetos ll the elow fomulted popetes e sed o the popetes of el umes d the e esult of the deftos of le veto opetos Poofs e esl pefomed the ede ) The ommuttve lw fo ddto: ) The ssotve lw fo ddto: ( ) ( ) ) The dstutve lws fo multplto ove ddto: λ ( ) λ λ λ µ ) λ µ ( Deomposto of Vetos to Compoets Retgul Othogol Bss ) et { } e the ut veto the postve deto of the - s veto { } e epessed s { } { } The veto s sd to e ss oe-dmesol spe of vetos 6

65 Vetos ) et { } d { } e two ut vetos the postve detos of the -s d -s espetvel veto } e epessed s { { } {} { } The s tht d e the ss vetos two-dmesol spe of vetos ) et { } { } d { } e thee mutull othogol ut vetos the postve detos of the Ctes oodte es veto { } e epessed s the le omto of the vetos d : } {} {} {} { Theefoe we ot the esoluto of t veto () lled the oodtes of the veto wth espet to ths ss 6 ove the othogol ss of vetos whee quttes d e

66 Vetos e Depedee of Vetos et K e vetos d λ λ K λ e umes The epesso of the fom λ λ K s lled le omto of the vetos K If thee ests o-tvl soluto of the homogeeous veto equto λ λ K () λ λ wth espet to λ λ K λ the t s sd tht { K } s the set of le depedet vetos Othewse f equto () hs ol the tvl soluto λ λ K λ the { K } s lled the set of le depedet vetos I othe wods the set of vetos s le depedet f oe of the vetos e epessed s le omto of the othe vetos of the set Fo ste f λ the ( λ K λ ) λ Theoem: ) two o-eo vetos e le depedet f d ol f the e olle ) thee o-eo vetos e le depedet f d ol f the e opl ) fou vetos thee-dmesol spe e le depedet Impott ote: The theoem sttes tht two o-olle vetos e le depedet d thee o-opl vetos e le depedet Poof: ) Two vetos d e le depedet f the equto λ λ hs o-eo soluto wth espet to λ d λ I ths se λ s the opposte veto of λ tht s d e olle vetos Hee two olle vetos e le depedet d two oolle vetos e le depedet 66

67 67 Vetos ) Cosde set of thee vetos { } I the oodte fom the veto equto λ λ λ e epessed s the homogeeous sstem of the followg le equtos: λ λ λ λ λ λ λ λ λ t fst let us ssume tht the vetos { } { } d { } e opl The thee ests oodte sstem whh Theefoe the ove homogeeous sstem s edued to the sstem of two le equtos wth thee uows λ λ d λ d hee hs o-eo soluto Thus set of thee opl vetos s le depedet ssume ow tht vetos d e o-opl le omto of vetos d s veto lg the sme ple s d Hee ot e epessed s le omto of d d so set of thee o-opl vetos s le depedet ) I se of fou vetos the equto λ λ λ λ s equvlet to the homogeeous sstem of thee le equtos wth fou uows λ λ λ d λ Suh sstem hs fte ume of solutos Hee set of fou vetos s le depedet set of le depedet vetos s lled ss the -dmeso spe of vetos Theefoe thee o-opl vetos fom the ss the thee-dmesol spe of vetos tht s veto d e epessed s le omto of the ss vetos: d d d d Ths fomul geeles the oept of the etgul veto ss { } to t set { } of o-opl vetos theedmesol spe The umes d d d d e lled the oodtes of d the ss of vetos d

68 Vetos Veto Bses et { } d { ~ ~ ~ } e two dffeet ses theedmesol spe of vetos B the theoem of le depedet vetos d d d d () d ~ ~ ~ ~ ~ d d d d ~ () fo t veto d I ode to fd eltos etwee the oodtes of d these ses we eed to esolve vetos ~ ~ ~ d to the ss vetos d : ~ ~ ~ Coeffets of the le omtos e the oodtes of the vetos ~ the ss of vetos d Susttutg these epessos eqult () d omg the sml tems we ot ~ ~ ~ d ( d d d 68 ) I vew of eqult () we get the tsfomto fomuls of the oodtes of veto fom oe ss to othe: ~ ~ ~ d d d d ~ ~ ~ d d d d ~ ~ ~ d d d d tsfomto of etgul ss otto of the oodte sstem s osdeed seto 7 Emple: et e gve the veto esoluto d 7 d the ss vetos ~ ~ { 8} d ~ { } {7 } To fd the oodtes of d the ss of ~ ~ ~ d we hve to solve the sstem of le equtos: ~ ~ ~ d d 7d ~ ~ ~ ~ ~ ~ 7 d d d d d d d ~ ~ ~ d 8d d

69 Sl Podut of Vetos Vetos ssume tht vetos d e gve the oodtes etgul oodte sstem: } d } { { The sl podut s ume tht equls the sum of the poduts of the oespodg oodtes: (6) The sl podut s lso ow s e o dot podut It s lso deoted s ( ) Theoem: If θ s the gle etwee vetos d the osθ (7) whee d e the legths of the vetos Poof: et us hoose etgul oodte sstem suh tht oth vetos d le the -ple; the -s s deted log the veto Se The theoem sttes tht d osθ we ot the desed esult P o P o os θ whee d e the ut vetos the detos of d oespodgl π If the os θ os whh mples the followg othogolt odto of the vetos } d }: { If the θ os θ d so 69 {

70 Vetos Theefoe the legth of the veto s epessed s pplg fomuls (6) d (7) we fd the ose of the gle etwee vetos d : osθ The most mpott ppltos of the sl podut e elted wth fdg the gle etwee vetos Popetes of the Sl Podut The elow popetes e sed o the defto of the sl podut The e esl poved the ede ) The sl podut s ommuttve: ) The sl podut s dstutve: ( ) ) If the sl podut of two o-eo vetos equls eo the the vetos e pepedul; d ve ves f two vetos e pepedul the the sl podut equls eo: Emples Emple : et { } { } d { } e thee ss vetos etgul Ctes oodte sstem The Emple : If { } { 7 } d θ s the gle etwee vetos d the ( ) 7 ( ) 7 8 7

71 os θ 7 6 Vetos Emple : Fd the gle etwee two vetos { } d { 7 } Soluto: ( ) Se the sl podut s equl to eo the vetos e othogol Emple : et p d q Smplf the sl podut of the vetos p d q Soluto: p q ( ) ( ) - Emple : Gve two sdes B d C of the tgle BC d the gle θ etwee these sde fd the thd sde of the tgle Soluto: et us deote B C d CB The ( ) osθ Deto Coses et α β d γ e the gles etwee ut veto u d the es of etgul oodte sstem The oses of these gles e lled the deto oses Theoem: I etgul oodte sstem the oodtes u u d u of ut veto u u u u } e equl to the deto { oses The theoem follows fom the defto of the sl podut The sl podut of the ut vetos u u u u } d { } e wtte { s u u d u u os α osα d so u osα 7

72 Vetos ewse u u os β d u u osγ whh equed to e poved B the defto of ut veto u u u u Theefoe os α os β os γ The deto oses of t veto e epessed s os α os β os γ Gve the vetos Veto Podut } d } etgul { oodte sstem the veto podut the fomul { s the veto whh s defed (8) whee d e the ut vetos of the etgul oodte ss The veto podut s lso ow s oss podut It s lso deoted s [ ] Epdg the detemt the fst ow we ot ) ( ) ( ) ( Theoem: et d e two o-pllel vetos The ) the veto s othogol to oth d ; ) the legth of s epessed the fomul sθ whee θ s the gle etwee d ; ) the set of vetos { } s ght-hded tplet s t s show the fgue elow 7

73 Vetos Poof: et the etgul oodte sstem e hose suh tht oth vetos d le the -ple d the -s s deted log The { } d { osθ sθ } Theefoe sθ osθ sθ Theefoe sθ d s deted log the -s whh s pepedul to the -ple Hee the theoem Popetes of the Veto Podut ) The veto podut s t-ommuttve: ) The veto podut s dstutve: ( ) ) The legth of the veto s equl to the e of the pllelogm wth det sdes d Cooll: The e of the tgle wth det sdes d s gve fomul 7

74 Vetos 7 ) The veto podut of two olle vetos equls eo Popetes ) d ) follow fom the popetes of detemts Ideed ) ( Popet ) follows fom the theoem of veto podut Popet ) s qute evdet Emples ) et { } { } d { } e thee ss vetos of the etgul Ctes oodte sstem B the defto of the veto podut Tht s ) et p d q Smplf the veto podut of the vetos p d q Soluto: ) ( ) ( - q p ) et BC e tgle wth the vetes t the pots ( ) B( ) d C( ) Fd the e of the tgle Soluto: Cosde the vetos d } B { } { C

75 Vetos 7 B the popetes of the veto podut Fd the veto podut: Theefoe 8 ) ( ) ( 6 Sl Tple Podut The sl podut d the veto podut m e omed to the sl tple podut (o med podut): ) ( ) ] ([ Theoem: Gve thee vetos } { } { d some etgul oodte sstem the sl tple podut s defed the fomul } { ) ( (9) Poof: Cg out the sl podut of the vetos ) ( ) ( ) ( d we ot ) ( ) ( ) ( ) ( Geomet Itepetto The solute vlue of the ume s the volume of pllelepped fomed the vetos ) ( d s t s show the fgue elow

76 Vetos Ideed the volume of the pllelepped s equl to the podut of the e of the se d the heght B the theoem of sl podut ( ) osϕ The qutt equls the e of the pllelogm d the podut osϕ equls the heght of the pllelepped Cooll : If thee vetos e opl the the sl tple podut s equl to eo Cooll : Fou pots B C d D le the sme ple f the sl tple podut (B C) D s equl to eo 6 Popetes of the Sl Tple Podut Cosde the sl tple podut ( ) ) B the popetes of the sl podut ( ) ( ) ) I vew of the popetes of detemts Theefoe ( ) ( ) 76 Se the ode of the dot d oss smols s megless the podut ( ) s smpl deoted Usg the popetes of detemts t s ot dffult to see tht

77 Theefoe ewse Vetos d so I vew of the theoem of le depedet vetos thee le depedet vetos e opl Hee The tple podut of o-eo vetos equls eo f d ol f the vetos e le depedet 6 Emples ) Deteme whethe the pots ( ) B ( ) C( ) d D( ) le o the sme ple Soluto: Jo the pot wth the othe pots to ot the vetos B { } C { 8} d D { } Fd the sl tple podut: 8 8 Theefoe the vetos le ple tht mes the gve pots le the sme ple ) Fd the volume V of the tethedo wth the vetes t the pots ( ) B ( ) C ( ) d D( ) Soluto: Cosde pllelepped whose det vetes e t the gve pots The volume of the pllelepped s equl to the solute vlue of the V p tple sl podut of the vetos B C d D The volume of the tethedo s gve the fomul V V p 6 77

78 Vetos Se B { } C { } d D { } we ot B C D Theefoe V 6 ) The tethedo s gve the vetes ( ) B ( ) C ( ) d D( ) Fd the heght fom the pot D to the se BC Soluto: I vew of the fomul V S h whee h s the heght fom the pot D we eed to ow the volume V of the tethedo d the e S of the se BC to fd h odg to Emple the volume of the tethedo equls The e of the tgle BC e foud ust sml w s Emple seto : B C B C B C Theefoe V h 78

79 79 Vetos 7 Tsfomto of Coodtes Ude Rotto of Coodte Sstem Cosde etgul Ctes oodte sstem et e { } e { } d e { } e othogol ut vetos of tht sstem: e e δ () whee δ s the Koee delt B otto of the oodte sstem we ot ew etgul oodte sstem et { } { } d e { } e othogol ut vetos of the ew oodte sstem tht s e e δ () B the theoem of deto oses the oodtes of the vetos e e d e the ss of vetos e e d e e the deto oses Deote the deto oses of the veto e (wth ) u u d u espetvel The e u e e u e u e u e e u e u e u e e u e u o shot fom e u e ( ) () `Equltes () d () mpl u u u u δ ( ) () Rewte equltes () d () the mt fom et U u e the mt of the deto oses If we todue the olum mtes E e d E e the T T E U E U U U U I T whee U s the tspose of mt U d I s the dett mt Note tht the tspose of U s the vese of U Theefoe we esl ot the fomul of the vese tsfomto (fom the old ss to the ew oe): E U E U E U U E T E U E U E

80 Vetos 8 Ths mt eqult s equvlet to the sstem of thee veto equltes: u e e ( ) Now osde the tsfomto of the oodtes of t veto veto e epessed s the le omto of ss vetos If e the oodtes of the old ss d e the oodtes of the ew ss the d d e e e e () e e e e () Theefoe e e I vew of eqult () we ot u e e u e e tht esults the fomuls of tsfomto of the oodtes: u u u u (6) ewse u u u u (6) Utl ow we hve tepeted the sl podut of vetos s sl qutt wthout foml poof osdeg ths poposto s the selfevdet tuth goous ustfto follows fom the elow theoem Theoem: The sl podut s vt ude otto of the oodte sstem Poof: Rell usg fomuls (6) we ot u u u u δ

81 7 Rotto of the Ple oud the -s Vetos Cosde ptul se of tsfomto of the etgul oodte sstem otto of the ple oud the s et θ e the gle of the otto d e the dus-veto of pot M The B the popetes of the sl podut osθ os( 9 θ ) sθ os( 9 θ ) sθ osθ Theefoe the sl poduts d e epessed espetvel s osθ sθ (7) d sθ osθ (7) ewse fomuls of the vese tsfomto follow fom the sl poduts d : osθ sθ (8) sθ osθ (8) Fomuls (7) (8) e ptul ses of geel fomuls (6) Emple: et θ Epess the qudt fom the old oodte sstem Soluto: ppl fomuls (8) tg to out tht s θ osθ : ( ) ( ) ( )( ) ( ) 8

82 ltl Geomet NYTIC GEOMETRY 6 Stght es 6 Equtos of es deto veto of stght le s veto pllel to the le odg to the postultes of geomet pot M d deto veto q deteme the stght le et M e t pot o the le The dffeee etwee the dus-vetos of the pots M d s veto the le tht s M q Two pllel vetos e popotol: t q () Ths veto eqult s lled the veto equto of the le t ume t s sd to e pmete ssume tht etgul Ctes oodte sstem s hose The equto () e wtte the oodte fom s the sstem of thee le equtos q t q t () q t whee d e ug oodtes of pot o the le Vetos d q e epeseted the oodtes: } { q { q q q} Equtos of le oodte fom () e lled the pmet equtos of le 8

83 ltl Geomet Solvg sstem () elmto of the pmete t we ot the ol equtos of le: q q q () If M ( ) d M( ) e two gve pots o le the the veto q } { og these pots seves s deto veto of the le Theefoe we get the followg equtos of le pssg though two gve pots: () Emples: ) et e le pssg though the pots M ( ) d M ( ) Che whethe the pot (7 ) le o the le Soluto: Usg () we get the equtos of : The oodtes of the pot stsf the equto: 7 d so s pot of the le ) Wte dow the ol equtos of the le pssg though the pot ( ) d eg pllel to the veto q { } Soluto: B equto () we ot Note tht smoll otto mes the equto 8

84 ltl Geomet 6 es Ple O the ple le s desed the le equto B C () If M ( ) s pot o the le the B C (6) Suttg dett (6) fom equto () we ot the equto of le pssg though the pot M ) : ( ( ) B( ) (6) The epesso o the left hd sde hs fom of the sl podut of the vetos { B} d }: { ( ) Theefoe the oeffets d B e tepeted geometll s the oodtes of veto the ple eg pepedul to the le The ol equto of le the ple hs fom q q whee q { q q} s deto veto of the le I the ple equto of le pssg though two gve pots M ( ) d M( ) s wtte s follows Sometmes t s helpful to epess stght-le equto the ple s (7) I ths se mples d mples 8

85 ltl Geomet Theefoe the quttes d e espetvel the -teept d the - teept of gph of the le Equto (7) s lled equto of le the teept fom le o the ple m e lso gve the equto the slopeteept fom whee s the -teept of gph of the le d s the slope of the le If M ( ) s pot o the le tht s the the pot slope equto of the le s ( ) Emples: ) le o the ple s gve the equto Fd: () two pots o the le; () the slope of the le; () the d teepts Soluto: () Settg we ot 8 If the Theefoe the pots P( 8) d Q( ) le o the le () 8 Theefoe the slope of the le s () The teept equls 8 The teept s the soluto of the equto tht s ) I the ple fd the equto of the le pssg though the pot M ( ) d eg pepedul to the veto N { } Soluto: Usg equto (6) we ot ( ) ( ) 7 8