Geometric Analogy and Products of Vectors in n Dimensions

Transcription

1 Adaces Lear Algebra & Matrx Theory Pblshed Ole March 0 ( Geometrc Aalogy ad Prodcts of Vectors Dmesos Leoardo Smal Morera UFOA Cetro Uerstáro de Volta Redoda Volta Redoda Brazl Emal: smalleoardo@terracombr leoardomorera@foaorgbr Receed Noember 0; resed December 0; accepted December 0 0 ABSTRACT The cross prodct Ecldea space IR s a operato whch two ectors are assocated to geerate a thrd ector also space IR Ths prodct ca be stded rewrtg ts basc eqatos a matrx strctre more specfcally terms of determats Sch a strctre allows extedg for aalogy the deas of the cross prodct for a type of the prodct of ectors hgher dmesos throgh the systematc crease of the mber of rows ad colms determats that costtte the eqatos So a -dmesoal space wth Ecldea orm we ca assocate ectors ad to obta a -th ector wth the same geometrc characterstcs of the prodct three dmesos Ths kd of operato s also a geometrc terpretato of the prodct defed by Eckma [] The same aaloges are also sefl the erfcato of algebrac propertes of sch prodcts based o kow propertes of determats Keywords: Cross Prodct; Space IR ; Determats; Geometrc Aalogy; Eckma s Prodct Itrodcto I the Ecldea space IR the cross prodct of two ectors ad s the ector desgated by the symbol ad defed for the followg codtos []: a) The orm of ector symbolzed for s ge for k () where k se beg the agle betwee the ectors ad b) The ector s perpedclar smltaeosly to the ectors ad : 0 () 0 () As a coseqece of b) s the ormal ector to the plae defed for the ectors ad (Fgre ) f these are learly depedet ectors Cosderg pqr the : px qy rz c 0 where c pa qa ra represets the eqato of the plae a Cartesa coordate system ( Aa a a s a pot IR ad A ) If ad are learly depedet ectors the 0 () where the symbol 0 represets the ll ector c) The ector s oreted relato to the ectors ad st as rght-haded coordate system the z-axs t s oreted relato to the x-axs ad y-axs d) The olme V of parallelepped defed for the ectors ad s the sqare of the mber (Fg re ): V (5) The eqaltes () () ad (5) are eqalet to those ge a Defto fod [] I ths paper t s show that t s possble throgh smple aaloges wth the case the space IR to exted the deas of the cross prodct to the space IR ad more geerally to the space IR The characterstcs of the cross prodct IR are mataed hgher dmesos Matrx Strctre of The tal reasog for the exteso of the deas of the cross prodct s the fact that ther basc expressos ca be represeted the form of determats I a orthogoal coordate system represetg the ectors ad terms of -tples ad the ector ca be obtaed startg from the deelopmet of the symbolc determat Copyrght 0 ScRes

2 L SIMAL MOREIRA Fgre [] s the ormal ector to the plae defed for the ectors ad cos k se 0 π (0) cos ad combg the Eqatos (9) ad (0) we obta () cos cos Eqato () wll be sed as startg pot for the aaloges deeloped the remag of ths work Fgre Parallelepped defed for the ectors ad [] eˆ eˆ eˆ (6) where eˆ 00 eˆ ˆ 00 e 00 are the ectors of orthoormal bass IR The deelopmet of the Eqa to (6) leads to the ector form: ˆ ˆ ê (7) e e ad the orm of ector s calclated wth the defto of Ecldea orm resltg a eqalet format to I Eqato () (8) (9) Exteso of the Cross Prodct to the Ecldea Space IR Cosder three ectors Ecldea space IR represeted terms of qadrples ad Let eˆ 000 eˆ 0 00 eˆ 000 ad eˆ 000 be the ectors of orthoormal bass IR It s possble to deelop a eqalet prodct to () throgh smple exteso of deas ad crease of dme- sos I space IR two ectors ad geerate a thrd ector whose orm s proportoal to t he prodct of the orms of the geeratg ectors beg the proportoalty costat related to the agle betwee ad I space IR three ectors ad geer- ate a forth ector whose orm s proportoal to the prodct of the orms of the geeratg ectors beg the proportoalty costat related to the agles betwee the ectors ad ad ad I symbolc terms ths prodct of ectors Ecldea space IR s obtaed from the deelopmet of the determat eˆ eˆ eˆ eˆ h () so that h k () wth cos cos k cos cos () cos cos Copyrght 0 ScRes

3 L SIMAL MOREIRA π ad the codtos The eqal sg the codtos o the agles ge () s stfed for the case of coplaar ectors I Eqato () cos represets the a gle betwee two of the geeratg ectors of h ad atrally cos cos so that k s the determat of a symmetrc matrx The eqalet space IR of Eqato () s (see the Eqato (5) below): The characterstcs of the prodct space IR are cosered for h space IR : a) The orm of h s proportoal to the prodct It s sffcet to deelop the determats Eqato (5) to erfy the detty b) The ector h s perpedcla r to each oe of the e ctors ad The term perpedclar shold be terpreted here as oly the sese that the scalar prodct h reslts ll PROOF: The elemets of the st row of the determat that represets the orm of h are the same ales as ther ow cofactors It s kow that the sm of the prodcts of the elemets of a row for the cofactors of the elemets correspodg of other row (er prodct) a determat reslts zero (Cachy s Determat Theorem) that s h h h 0 It s also oted that h s the ormal ector to the hyperplae that cotas ad Beg h heˆ heˆ ˆ ˆ he he the :hx hx hx hx c 0 where c ha ha ha ha represets the Cartesa eqato of hyperplae ( Aa a a a s a pot IR ad A ) c) The s e ectors ad st as the ector ê relato to ê ê ad ê d) The cotet of parallelotope defed for the ectors ad h s the sqare of mber h PROOF: Wth effect the determat to the left Eqato (5) represets the mber h I ths way h s the determat whose rows are formed by the ectors h ad represetg the cotet of parallelotope (-parallelepped) that has the for ectors as edges learly depedets [] Prodct of Vectors Ecldea space IR Cosder ectors Ecldea space seted terms of -tples sch that The prodct H IR repre- space IR s a ector perpedclar smltaeosly to all the ad whose orm s ge by the formla H K (6) wth cos cos cos cos K cos cos (7) It s obsered that ths form s eqalet to the prodcts of ectors defed by [] ad cted [56] amely (sg the same symbols as [6]) that a cross prodct satsfes the axoms: (A) P a a 0 r a r (A) Pa a det r a a where a a a These prelmary deftos ca be formalzed startg from the followg proposto PROPOSITION: Let ectors be space IR wth er prodct ad Ecldea orm Cosder also that the ectors are represeted by -tples sch that ector h oreted relato to th Beg cos the agle betwee the -th ector ad the -th ector the followg eqalty s tre (see the Eqato (8) below): cos cos cos cos cos cos (5) Copyrght 0 ScRes

4 L SIMAL MOREIRA cos cos cos cos cos cos cos (8) PROOF: Cosder t ectors space IR wth er prodct ad Ecldea orm Cosder also that each represets the t ector the same drecto of ge the Eqato (8) so that (9) If the t ectors are represeted by -tples sch that beg the er prodct betwee the -th t ector ad the -th t ector ca be groped based o the propertes preseted (A) the compoets of the followg detty whch s tre for ales of : (0) Startg from Eqato (0) Eqato (8) ca be demostrated Wth effect mltplyg both members of (0) for the determat to the left wll hae ther rows orderly ad approprately mltpled by each oe of ad sce s obtaed the correspodg determat of Eqato (8) Represetg for coeece k k k () Copyrght 0 ScRes

5 L SIMAL MOREIRA 5 we hae that: k k k k () I relato to the determat to the rght Eqato (0) t s sffcet to obsere that therefore that s: cos cos cos cos cos cos cos c os Wth sch cosderatos t s demostrated that k k cos cos cos cos cos cos cos () () ad the sqare root of Eqato () shows that Eqato (8) s tre Eqato (8) s the eqalet -dmesoal of the Eqatos () ad (5) aldatg the exteso of cross prodct The geometrc propertes of H are cosered dmesos: a) The orm of H s proportoal to the prodct beg the proportoalty costat K assocated to the agles betwee the ectors PROOF: The proof cossts of the ow demostrato of the Eqato (8) b) The ector H s perpedclar to each oe of the ectors PROOF: The elemets of the st row of the determat that represets the orm of H are the same ales as ther ow cofactors I agreemet wth Cachy s Determat Theorem the sm of the prodcts of the elemets of a row for the cofactors of the elemets correspodg of aother row (er prodct) a determat reslts zero that s H H H 0 It s also oted that H s the ormal ector to the hyperplae that cotas Beg H H ˆ ˆ ˆ ehe He the : Hx Hx Hx C0 where C HaHa Ha represets the Cartesa eqato of hyperplae ( A a a a s a pot IR ad A ) c) The ector H s oreted relato to the ectors st as the ector eˆ s oreted relato to eˆ eˆ eˆ d) The cotet of parallelotope defed for the ectors ad H s the sqare of mber H PROOF: The determat to the left Eqato (8) represets the mber H I ths way H s the determat whose rows are formed by the ectors H represetg the cotet of parallelotope (-parallelepped) that has the ectors as edges learly depedets [] 5 Coclsos stace: (C) If w s the www 0 ; a) b) c) d) ww w The possblty to represet the eqatos of the def- to of cross prodct the space IR terms of determats allows the exteso of the cocept of the prodct of ec tors for hgher dmesos systematcally creasg rows ad colms to the determats Throgh basc propertes of determats t s show that the characterstcs of th e cross prodct are cosered dmesos for ay ale of sce sch propertes are ot modfed by the cremet or decrease of rows ad colms to these determats Other geometrc propertes ca be erfed as the relato shp betwee the cross prodct ad area becase st as the mber s related to areas of tragles ad parallelograms the mber H s related to cotets of smplex ad parallelotopes a eqalet way to Cayley-Meger determat [78] Althogh ths work has ge emphass to the geometrc propertes of the prodct of ectors the space IR t drectly shows that ther algebrac propertes are also smlar to those ald oes space IR for ay ector space IR for 0w w 0; ww 0 0; 0 f ay of ectors w s the ll ector (C) The posto chage amog two ectors the prodct W ww w reslts the ector W (C) If w s ay ector space IR for ad a IR the a) aw ww waw w ; b) aw w w aw w w These ad other algebrac propertes cldg the dstrbte property of the prodct relato to the sm of Copyrght 0 ScRes

6 6 L SIMAL MOREIRA ectors are erfed easly by the applcato of the coeet rles o determats to the matrx strctre of prodct of ectors The aaloges deeloped appear stll for the possblty of ew extesos assocated to the cocept of prodcts of ectors sch as eetal deelopmets that are related to a type of eqalet -dmesoal of the cocept of crl for example REFERENCES [] B Eckma Stet ge Lösge Learer Glechgssys- teme Commetar Mathematc Heletc Vol 5 9 pp 8-9 do:0007/bf [] N Efmo Elemetos de Geometra Aalítca Cltra Bras lera São Palo 97 [] A Eldqe Vector Cross Prodcts Talk Preseted at the Semaro Rbo de Fraca of the Uersdad de Zaragoza o Aprl 00 [] S Lpschtz ad M Lpso Álgebra Lear Bookma Porto Alegre 008 [5] R Brow ad A Gray Vector Cross Prodcts Commetar Mathematc Heletc Vol 967 pp - 6 do:0007/bf0568 [6] A Gray Vector Cross Prodcts o Mafolds Uersty of Marylad College Park 968 [7] P Grtzma ad V Klee O the Complexty of Some Basc Problems Comptatoal Coexty II Volme ad Mxed Volmes I: T Bsztrczky P McMffe R Scheder ad A W Wess Eds Polytopes: Abstract Coex ad Comptatoal Klwer Dordrecht 99 p 9 [8] D M Y Sommerlle A Itrodcto to the Geometry of Dmesos Doer New York 958 p Copyrght 0 ScRes