Coordinate Transformations

Size: px

Start display at page:

Download "Coordinate Transformations"

Cody Cross
6 years ago
Views:

1 Coll of E Copt Scc Mchcl E Dptt Nots o E lss Rvs pl 6, Istcto: L Ctto Coot Tsfotos Itocto W wt to c ot o lss lttv coot ssts. Most stts hv lt wth pol sphcl coot ssts. I ths ots, w wt to t ths oto of fft coot ssts to cos bt coot ssts. Ths pps th w fo th costo of fftl qtos ppl to l os sch s thos s ft-lt copt pos. H w focs o th coot tsfotos q to covt th fftl qtos, oll pss Cts coot ssts to oth ssts. Notto fo fft coot ssts Th l lss of coot tsfotos sll stts wth th qtos Cts bss,, sps of tsfoto of l lttv coot sst,,. Ths s sots pst s tsfoto fo Cts sst,, to th solss sst,,. Th ltt fo of th tsfoto llows th s of copct otto, toc blow, ow s plct sto ov pt cs. Th ts of t th w coot sst s th ts of f th ppopt tsfotos,,,,,,,,. I th cl sbscpt otto, ths tsfotos bco,,,,,,,,, Ths th tsfotos c b copctl wtt vcto otto:. I cl lss of copl ssts w hv to fo sh tht fts th bos of th sst b l. I sch css,,, th copttol coots whch tpcll ft to spl wh,,. Th vls of th copttol coots occ t th phscl bos of th t b l. Ths copttol coots th bco th pt vbls th qtos. Ths w hv to tsfo th fftl qtos w l fo th Cts coot sst to th s of,, s th pt vbls. I th scsso blow w pst l w to o ths tsfoto. Th tsfoto of th fftl qtos qs foto bot tsfoto of th spc vtvs. Th bsc ltos o th spc vtvs fo fo th qto fo th totl fftl of o w coot,, wh,,. Thos bsc qtos pss th fct tht fftl ch of th coots th ol coot sst c cs fftl ch o of th coots. Th l qto fo s v blow. o,, [] Eqto [] s wtt th ws. Th fst fo shows ll ts th qto. Th sco fo ots tht th th ts o th fst qto sl c b s s of th spt ts s sto. Th fl fo of qto [] s sl to th c E Ml Co Pho: E-l: lctto@cs. 848 :

2 Coot tsfotos L. S. Ctto, pl 6, P sco fo, cpt tht th sto s s ss. Ths s shoth otto to splf wt sch qtos. I ths shoth, th s pl sto ov th ts wth th pt. Ths s ow s th Est sto covto. W wll s ths pocll to t s to wt sch qtos. Th fl,, st bfo th qto b ppls to ll th qtos fo ; t s s tht th qto fo ppls fo th th fft vls of. I th of ths ots w wll s oft wt ts fll to s who ot fl wth ths covto tht w ctll cos svl ts b th pl sto. If w loo t th vs pobl of t th fftl chs o ol coot sst,,, fo fftl chs th,, coot sst, w wol hv th follow lo of qto []. o [] W c wt both qtos [] [] s t qtos to show tht th ptl vtvs, lt to ch oth s copots of vs t. I t fo, qto [] bcos. [] Covt qto [] to t fo vs th follow slt. [4] Eqtos [] [4] c ol b coct f th two th-b-th tcs tht pp ths qtos vss of ch oth. Tht s, th ptl vtvs lt b th follow t vso.

3 Coot tsfotos L. S. Ctto, pl 6, P [5] If t,, s th vs of t,, th copots of b v b qto [6]. I tht qto, M, ots th o tt whch s f s follows. If s -b- t, t hs o tts, M, whch th tts of th - b - tcs fo f ow col lt fo th ol t. Th o tt s s to f th cofcto, - M. Th copots of th vs t f ts of ths cofcto th tt of th ol t,., Dt M Dt b o [6] Th tt of th t o th ht h s of qto [5] s ow s th cob tt. Th sl pso fo tt vs th follow psso fo. Dt [7] Us qto [6] fo th copots of th vs t, wth th tt th oto st to, w f th follow ltoshps btw th vl t copots qto [5]. Ths vtvs cll th tc coffcts fo th tsfoto. I th qtos blow w wt ths coffcts both th l fo wth cl sbscpts s th,,,, otto. Th fl t ch qto s lttv otto fo ptl vtvs. o pl, s shoth fo th ptl vtv. W c vw qtos [5] [6] s follows. W t to f th coffcts of th vs t, b. Eqto [5] shows tht ths copots v b th qto b. I.., th ow,, s th to th col s th oto. cco to qto [6], ft tpt th tt s th cob,, th w c wt b. t s th cofcto of th t ow col of th cob. o qto [5], w s tht th t ths

4 Coot tsfotos L. S. Ctto, pl 6, P 4 o [8] o [9] o [] o [] o [] o [] o [4] o [5] o [6]] Th spl ltoshps fo two-sol coot ssts c b fo fo ths qtos b co tht sch coots, th s o vto th th so. Ths s tht th s o vto of o wth. Ths ll vtv of wth spct to o. W st th vtv to of qtos [8] to [6] fo two-sol ssts. Ths s qvlt to ss coot tsfoto of fo ths covso. Th slts of covt qtos [8], [9], [] [] to two-sol fos show blow o [7] posto s. Ths w c th follow l sttt bot th slts show qtos [8] to [6]: qls th cofcto of v b th cob,.

5 Coot tsfotos L. S. Ctto, pl 6, P 5 o o [8] [9] o [] o th two sol cs, th cob hs th spl fo of two-b-two tt. [] Not tht qto [6] s coct wh w covt fo th sos to two b stt. Th lft h s,, s ql to o. Th ts bcs o th lft-h s st th fto of th cob,, fo th two-sol cs. Ths both ss of qto [6] ql to o th two-sol cs. Tsfo fftl qtos W ow to tsfo th vos vcto optos fo Cts coots to o bt coot sst. W b wth th vc bcs ths s tsfoto of fst vtvs. Sbsqtl w wll cos th Lplc whch qs tsfoto of sco vtvs. s sl w wll sco vtvs s fst vtvs of fst vtvs b bl to ppl th slts of th fst-vtv tsfotos to th slts fo sco vtvs. Tsfo th vc Th vc of vcto wth Cts copots,,, th,, coot ctos h pss s,, s wtt s follows. Th sco fo ss th pl sto ov th pt,. v o v [] I copttol fl cs qtos, th covcto ts, wth ρ φ, v b ths vc psso. To c ot th tsfoto fo th vc, w co ch Cts coot c p o ll th oth coots. cs of ths, ch of th tsfo coots c b flct s ch of th ol coots. W c flct ths pc b wt th follow qto to covt fst vtvs o Cts coot sst wth spct to Cts coot, to fst vtvs w coot sst wh th coot vbls cll,, o,, o l.

6 Coot tsfotos L. S. Ctto, pl 6, P 6,, o [] Th sco fo of ths qto hs pl sto ov th pt. W hv to ppl ths qto to ch of th th ts o vc qto []. v [4] W c splf th b of ts tht w hv to wt b s th pl sto ov pt cs of th sto covto. H w pt two cs, whch llows s to wt ll ts o th lft of qto [4] th follow copct otto. Dv [5] Th covso of ths fo to o sfl slt os ot follow obvos pth. Th tl stp th covso s o b fst ltpl b th cob of th tsfoto,, ppl th ch l of fftto to wt th slt X ts s X X. Ths vs th follow slt. X X X Dv [6] W cot to hv two pt cs tht pl sto ov both. W c show tht th fl t, ltpl b, s o fo ch vl of b s th tc coffct ltoshps qtos [8] to [6]. W fst t th follow slt fo, s qtos [8], [], [4]. [7] C ot th ct ffttos vs th cobto of, sco-o ptl vtvs show blow. Ech of ths vtvs occs two ts, oc wth pls s oc wth s s. Th o of fftto s lso fft, bt sco o vtvs th s lss of th o of fftto. ltt blow th t wth pls o s s cts th tch ts tht ccl. o pl, th t lbl hs pls s th qto tht ccls th t lbl -.

7 Coot tsfotos L. S. Ctto, pl 6, P 7 C E D E D C [8] Ths shows tht th t qto [45] s o wh. Th poof tht ths t s o fo follows th s ppoch s bov s lft s cs fo th tst. Wth ll ths ts o, qto [45] vs th follow slt fo th tsfo covcto ts. Dv Dv [9] W c f th copots of th fftto b th w coot sst s follows. G [] Wth ths fto, th vc o w coot sst, wth th w copots G, bcos G Dv [] Tsfo th Lplc W c t ths slt to v fo fo th Lplc opto th w coot sst. Th Lplc c b vw s th vc of t. I Cts coots th Lplc s wtt s follows. Dv Dv [] H w hv s,, to pst th sl t vctos,,. W s tht th Lplc psts th vc of vcto whos copots. Howv,

8 Coot tsfotos L. S. Ctto, pl 6, P 8 w hv st fo psso fo th vc o w coot sst b th cobto of qtos [] []. ppl qto [] wth vs. G [] W wt th vtvs of wth spct to o w coot sst. To o ths w s th l ltoshp fo ptl vtvs tht vs s vtvs of th w coot sst; w c show ll ts o s th sto covto. [4] W c cob qtos [] [4] to t fto of G tht volvs pl sto ov th pt cs. I qto [5] w show ll ts tht slt fo th pl sto ov ths two pt cs. G [5] Wth ths fto fo G, w c s qto [] to wt o Lplc wth th pl sto ov th. G [6] Eqto [6] hs th pt cs,, whch pl sto ov ll possbl vls of ch. Ths vs totl of 7 ts qto [6]. Th th plct ts fo,, show blow. Ech of ths ts hs pl sto ov th pt cs. [7] oth vw of qto [6], show blow, cots ll th ts fo. Th ts fo lft s pl sto ov.

9 Coot tsfotos L. S. Ctto, pl 6, P 9 [8] Eqto [6] povs th ost cophsv fo fo th Lplc bt coot sst. W c ppl t to th splst pl of clcl-coot ssts wh cosθ, sθ. I o l coot sst, th Cts coots fo fo th w coots b th follow fo of ths tsfotos: cos, s. Th vtvs fo ths sst wtt blow. cos s s cos [9] W s qto [7] to copt s ths vtvs. [ ][ ][ ] [ ][ ][ ] [ ][ ][ ] [][ ][] [ ][ ][] [ ][][] [ ] s cos cos s s cos s s s cos cos [4] W ow hv to s qtos [8] to [6] to copt th vtvs fo th vtvs fo qto [5] th cob. [ ] cos cos [4] [ ] s s [4]

10 Coot tsfotos L. S. Ctto, pl 6, P [ ] cos s [4] [ ] s s [44] [ ] cos cos [45] [ ] s cos [46] [ ] cos s [47] [ ] cos s [48] [ ] s s cos cos [49]] fo sbsttt ths vtvs to qto [6] w ot tht w c wt qto [6] s follows, f s th poct of two fft ptl vtvs wth spct to s ov ll th vls of. wh [5] W c wt th plct fto of wthot th pl sto s follows. Not tht. [5] Us th vtvs qtos [4] to [49] th slt tht, w c wt th fcto fo qto [5]. s cos [5]

11 Coot tsfotos L. S. Ctto, pl 6, P cos s s cos [5] s cos [54] cos s [55] cos s [56] [57] W s tht ll th vls of o lss. Ths wll lws b th cs fo othool coot sst. o sch sst w c wt qto [5] to st ll ts wh to o. [58] Us th vls of,,, fo qtos [5], [55], [57], th slt fo qto [4] tht., w c wt o Lplc fo clcl pol coots. [59] Sc th th coot ctos pt w c ov th ts fo th vtvs fll s th covtol, θ, coots to v th fl slt fo th Lplc clcl coots. θ [6] Lplc s qto wth vbl popts s ot bov, th Lplc c b vw s th vc of t. Tpcll th t ts so phscl coffct, Γ, psts fl t. If th phscl coffct s costt, w c b t ots th ot vc opto t wth oth ts th qto. Howv, f Γ s ot costt, w hv to lv t s th ot vc opto. I ths cs, w plc qto[ ] wth th follow qto Cts coots.

12 Coot tsfotos L. S. Ctto, pl 6, P Γ Γ Dv Dv [6] H w hv s,, to pst th sl t vctos,,. W s tht th Lplc psts th vc of vcto whos copots. Howv, If w h cl ths G coffct o ol vto w wol hv obt th follow slt plc of qtos [6] [5]. Dv Γ Γ Γ [6] Vctos, s vols th w coots Posto vcto fftl lth To stt o costos of vctos, s, vols l coot sst, w cos posto vcto,, tht s f Cts coot spc s follows wth t vctos,. Th posto vcto s f s follows Cts sst. W s th,, o,, s o Cts coots.. [6] Th vtv of th posto vcto, wth spct to ptcl w coot,, s v b qto [64], whch s lso s to f th bs vcto,, th w coot sst.,, [64] I th lst psso, w s th sto covto ov th pt. Th th bs vctos f qto [64] th qvlt of th sl bs vcto tht w hv o Cts coot sst. Howv ths ll ot ol vctos. W c s ths bs vctos to copt th fftl vcto lth,, lo pth o w coot sst. [65] I th two l ts of qto [65] w hv s th sto covto. I ft qtos w wll s ths covto th sto ov pt cs wthot fth cot. W c wt lt lth Cts spc, s, s th t of th ot poct of wth tslf. s [66 Not tht ll th ts volv cs hv both cs pt. Ths w s ov both cs w hv ts ths css.

13 Coot tsfotos L. S. Ctto, pl 6, P Mtc coffcts Th ot poct of two bs vctos, s f, o of th copots of qtt ow s th tc tso. o th fto of qto [64], w c wt s follows. H w s th follow qto tht ss th fct tht th bs vctos th Cts coot sst t vctos whch tll ppcl. Sch sst of vctos s cll othool: δ. δ [67] o pl, clcl coot sst, cos, s. W hv th follow ptl vtvs. cos s s cos [68] Sbsttt ths vtvs to qto [67], llows s to copt so of th copots fo th clcl coot sst. [ ] [ ] s s cos s cos s s cos [69] Th, q, off-ol ts, c both b show to b o. Th off ol ts,,, s to b stc b th bsc fo of qto [68]. Ths ts wll lso b o. Wh th tc tso hs o fo ll ts off-ol ts, th slt coot sst s othool. I othool sst, ch bs vcto s ppcl to th oth two bs vctos t ll pots th coot sst. Th fftl pth lth v b qto [66], whch w s to f w t, fo othool ssts ol h. h h h h s [7]

14 Coot tsfotos L. S. Ctto, pl 6, P 4 I th qto [69] pl of clcl coots, w h h h, h. Ths th th ts qto [7], θ. W s tht h ltpls th fftl coot, θ, slts lth. Ths s l slt fo h coffct; ths coffct s fcto tht ts fftl coot cto covts t to phscl lth. Ths fcto lso pps optos o vcto copots fo othool ssts. Ths fctos sll wtt ts of Cts coots,, b th follow qtos, tht cobto of qtos [7] [67]. h h h [7] Dfftl Now tht w hv psso fo th fftl lth o w coot sst, w c v qtos fo fftl s vols. o qto [66] w s tht th lth of pth lo whch ol o coot, s, chs, s v b th qto o sto t; th vcto pstto of ths pth s. To t fftl fo two fftl pth lths, w t th vcto coss poct of ths two fftl lths. Th vcto coss poct vs th poct of two ppcl copots of th fftl pth lths to clclt fftl, S.,,,, cclc o sto S [7] Th vcto tht slts fo th coss poct s th pls o s coot cto p o whch cto th sfc s fc. Th oto tht,, cclc s tht w s ol th follow th cobtos,,,,,, o,,. I o to copt th t of th sfc, w to copt th t of th vcto coss poct. To obt sfl slt fo ths fto, w to s th follow vcto tt. C D D C D C [7] Us C D, vs th follow slt fo th coss poct of bs vctos. [5] Wth ths psso, w c wt th t of th fftl sfc cto s follows.,,,, cclc sto o S [74]

15 Coot tsfotos L. S. Ctto, pl 6, P 5 Dfftl vol W c t t fftl vol lt b t th vcto ot poct of fftl lt qto [] th fftl lth lt ol to th. Ths vs th fftl vol lt b th follow qto. V o sto,, cclc,, [75] st s w fo th fftl lt, w lso s th t of th vcto t th vol lt qto. Ths qs tht w f th t. To stt ths, w th follow vcto tt. [ C ] [ C C ] [ C ] [76] W c s th tt qto [4] to sbsttt fo th t C C. W c lso s th follow tt to sbsttt fo th C t. C C C [77] Sc w hv ol th bss vctos, w wll s th follow bs vctos fo qto [76] qto [77]:,, C. M ths sbstttos co tht th ot poct, th tc coffct, vs th follow slt. [ ] [ ] I qto [78], w hv s of th st ltoshp fo th tc tso copots,, obt th th l. W s tht ths fl l s st th qto fo th tt of. If w wt ths tt s, w hv th follow slt fo th vol lt. [78] V [] Dt Th pp cots poof tht th vl of s th s s th vl of th cob tt qto [7]. If w t to o pvos pl of clcl coot ssts fo whch,,, th vl of s spl th poct of th ol ts whch s ql to o th covtol otto. o ths sst, qto [] fo V vs th sl slt fo th fftl vol clcl coot sst, V θ. Ecs: o th sphcl pol sst th th coots, th stc fo th o to pot o sph,, th cotclocws l o th - pl fo th s to th pocto of th coot o th - pl, th l fo th vtcl s to th l fo th o to th pot. Ths coots o covtoll cll, θ, φ. Th tsfoto qtos fo Cts coots,, to sphcl pol coots v b th follow qtos:, t - /,

16 Coot tsfotos L. S. Ctto, pl 6, P 6 t - [ / ]. Th vs tsfoto to obt Cts coots fo sphcl pol coots s: cos s, s s, cos. ll copots of th tc tso fo ths tsfoto. Vf tht ths s othool coot sst. Wht th th possbl fftl s fo ths sst? Wht s th vol lt fo ths sst? Vcto copots l coot ssts Th splst vcto to cos l coot sst s th vloct vcto, v, whos copots th vtvs of th coots wth spct to t. W c f th vloct copot ptcl cto,, b th sbol v. Th fto of v th bt coot sst, ts ltoshp to th Cts coot sst s show blow, wh w hv s qto [] o [] fo th coot tsfoto, sbsttt th otto of fo th Cts coots. v t t t t t [] W s tht th ts /t o th ht-h s of qto [] st th vloct copots th Cts coot sst. I to, th s o ptcl so to ss tht th ol sst s Cts, w col qll wll s th otto fo th lttv coot sst th otto v fo th vloct copots tht sst. Ths vs th follow qto fo th tsfoto of vloct copots fo o coot sst to oth. v v v v t t t t v [] Ths tsfoto qto fo copots of th vloct vcto c b cotst wth th tsfoto qto fo th copots of th t vcto. Th t of scl, ; wtt s, hs th follow qto Cts coots. [] If w ot o copot of ths vcto s, w c wt ths copot ts coot tsfoto to w sst [4] If w cop qto [4] fo th tsfoto of th copots of t vcto wth qto [] fo th tsfoto of th copots of vloct vcto, w s tht th s sbtl ffc th qtos. o tsfo th t vcto fo th ol ā copots to th w copots, th ptl vtvs of th coots hv th to. o th tsfoto of th vloct copots fo th ol coot sst, v, to th v copots of th w sst, th ol coots,, pp th oto. It ths pps tht w hv two fft qtos fo th tsfoto of vcto.

17 Coot tsfotos L. S. Ctto, pl 6, P 7 Wht w hv, fct, s two fft s of vctos f b th tsfoto qtos. vcto tht s tsfo fo o coot to sst to oth s qto [] s cll cotvt vcto. O tht tsfos cco to qto [4] s cll covt vcto. Yo c b ths s b f o b tht covt vctos hv tsfoto ltos fo vcto copots whch th ol coots colloct wth th ol vcto copots th to of th tsfoto. Th tsfoto ltos fo cotvt vctos hv th ol coots th ol vcto copots loct th oppost loctos ol vcto copots th to ol coots th oto. I ccoc wth ths s w cll th vloct cotvt vcto th t covt vcto. lthoh th tll two tps of vctos, cco to th tsfoto ltoshps, ths ffcs spp fo othool coot sst. I to, o c pss covt vcto b ts cotvt copots vc vs. Th covt vcto copots pst th copots lo th coot ls. Th cotvt copots pst th copots lo ol to pl whch th coot vl s costt. vcto, sch s vloct, lws hs th s t cto t v locto flow. Th ol th tht vs fft coot ssts s th s whch w choos to pst th vcto. I othool sst, ol o choc of coot sst chs th pstto of th vcto. I oothool sst w choos ot ol th coot sst, bt lso whth w wt to pst th vcto b ts covt o cotvt copots. lthoh ch of th ol wo o bo ftt coot ssts s fft psttos of vloct copots, ost ct ppochs s folto. Th coot sst s oothool, bt w s Cts vcto copots. Ths s l s -θ- coot sst, bt lv th vloct copots s v, v, v. Ths s ot ws choc, bt t s possbl. Wh w l wth copl bo-ftt coot ssts, th s of Cts vcto copots os poc spl slts fo th CD clcltos. pp Poof tht / If w wt th tpcl lt th cob qto [7] tt s L, w s tht ths lt c b pss b th follow qto. H w cos th tsfoto fo Cts coots,,,, to w coot sst,,. L [-] W c wt th vl of th-b-th tt s th ptto opto,, whch s f s follows:, s o f two of ts cs th s; t s f th cs v ptto of t s f th cs o ptto of. ptto of s o o v f o o v b of chs s q to t fo to th v ptto. o pl qs chs s v;,, q o ch o; q two chs v. ll th vls of show th tbl blow

18 Coot tsfotos L. S. Ctto, pl 6, P 8 Th tbl shows tht ol s of th vls oo. Us ths opto th sto covto ov pt cs vs th follow fol fo th-b-th tt Dt [-] lthoh th totl of 7 possbl ts ths sto, ll bt s of th hv o vl fo ; th of th oo vls th - whch wll v s th sl fol fo th pso of th-b-th tt. qvlt fol vss th sbscpts of th ts ths qto. Dt [-] Us ths pssos, w c f th cob tt s follows L L L L L L [-4] W c t sbsttt qto [-] fo L to qto [-4]. [-5] To copt, wth vw to cop t to th tt,, w wt th two fctos wth th two fft fos of qto [], b cfl to s two fft sts of cs to ot tht ch tt hs spt pso. Ths vs th follow slt fo. o o o o [] If w p th o ptto opto ths qto w t th follow slt. [f] Nt, w s qto [b] to wt th tt of th tc tso,. []

19 Coot tsfotos L. S. Ctto, pl 6, P 9 o qto [8], w c wt th fto of, wthot th pl sto ov th pt. [h] W c cob qtos [f] [h] to obt th follow psso fo th tt, : [] Mltpl ot th ts pthss vs th follow qto fo. [] W wt to show tht qto [f] fo vs th s slt s qto [] fo. W s ths qtos ss of ts tht hv th s l fo. Ech t s th poct of s ptl vtvs tht c b pss s. tho th otos of th ptl vtvs ch t hv th s coo cs,,,,,. Howv, th

20 Coot tsfotos L. S. Ctto, pl 6, P cs th to of th ptl vtvs qtos [f] [] ot th s. lthoh both qtos lt th cs to, o, qto [f] hs ch occ ctl two ts, whl qto [] hs ts wh o o two cs ot b pst th to of th ptl vtv. Th ffc th ptl-vtv tos btw qtos [f] [], s wll s th l b of ts qto [], ssts tht so ts qto [] wll ccl wh th ptto opto s ppl. W c show, b o pl, tht ths wll lws b th cs wh o of th cs s ss fo th to ts qto []. E th tpcl t wh th ol two cs th to. Th s of ll s ts t b th ptto opto ths cs s show blow. Itcl ts occ wth both pls s s ct b cptl ltts wth pls o s s, blow th t. C C [] W s tht ths ts ll ccl. lthoh w hv ths ostto th cs wh th fst fo cs w th s th lst two cs w th s, w wol obt th s slt, lss of th locto of th fft cs. W ths cocl tht ll ts qto [h-b], whch o ot hv ll th cs th to wll vsh wh th ptto opto s ppl. Elt ll sch ts fo qto [] vs th follow slt. [l] W wt to hv th s slt fo qtos [f] [l]. It s ot ppt tht ths th s. I fct, th ts [l] ll hv postv ss, bt hlf th ts qto [f] hv tv ss. To show tht ths th s q fth t. W stt b qto [f] to t loo o l [l].

21 Coot tsfotos L. S. Ctto, pl 6, P [] W c lso qto [l] s follows [] Now w wt to show tht qtos [] [] v th s slt. W s tht th fst t ch qto s th s, bt th ffcs th oth ts. Sc th sto cs bt, w c ch cs. Howv, wh w pt cs, w hv to ch th s of th ptto opto. W c o ths fo ch t, cpt th fst t qto [], whch qs o ofcto. Stt wth th sco t th fst ow of qto [] w c swp th cs to v. [o] Th t o th ht of qto [o] s th s s th lst t qto []. I sl w w c swp th cs th fst t th sco ow of qto [] to show tht ths s th s s th sco t th sco ow of qto []. [] W c pt ths pocss fo th th ts ch qto tl w show, o t-b-t bss, tht th qtos fo th s. Ths coplts th poof tht both qtts th s.

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,