Vectors and vector calculus

Transcription

1 Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus Vecrs in w nd hree dimensins Free ecrs: A ecr is free if i is cnsidered e he sme ecr under rnslin PQ is free ecr ecuse i cn e repsiined s shwn in he fllwing digrm P Q Surcin is dne dding he secnd ecr in reerse direcin he firs ecr c = - Liner dependence nd independence Vecrs nd z re linerl dependen if here eis nnzer sclrs p q nd r such h p + q + rz = If p + q + rz = nl when p = q = r = hen nd z re linerl independen A psiin ecr is n free ecr ecuse i cnn e rnsled I lws srs frm reference pin O he rigin Emple OP nd OQ shwn elw re psiin ecrs Q w z P O Since w + + z = ie w + + z = Therefre w nd z re linerl dependen Ms ecrs cn e dded r surced if he represen he sme quni Emple Three nn-cplnr ecrs re linerl independen Ecepins: Addiin f w psiin ecrs is undefined A displcemen ecr cn e dded psiin ecr gie new psiin ecr Surcin f psiin ecrs is defined s displcemen A picl emple is he cse where he hree ecrs re perpendiculr ech her 4 Frce ecrs cing n differen jecs cnn e dded Addiin f ecrs is crried u puing he hed f ne he il f he her The rder is irrelen c = + Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus

2 Emple Gien = unis nrh nd = 4 unis es find ecr c which is linerl dependen n nd Find nher ecr d which is ls linerl dependen n nd A pssile ecr c is c = + ecuse c = ; pssile ecr d is d = ecuse d = Resluin f ecr in recngulr cmpnens A ecr is resled in recngulr cmpnens when i is epressed in erms f i j nd k eg r = li + mj + nk The mgniude f r is r = r = l + m + n Emple 4 Gien h ecrs p q nd r re linerl independen nd (p + q r) = ( + 6)p + ( + )q 9r find he lue(s) f (p + q r) = ( + 6)p + ( + )q 9r ( 6) p + ( + ) q ( 9) r = Since p q nd r re linerl independen 6 = + = ( 9) = ie ( + )( ) = ( + )( ) = ( + )( ) = = is he nl lue h sisfies ll hree equins Uni ecrs An ecr wih mgniude f is clled uni ecr A uni ecr in he direcin f ecr s is lelled s ŝ I is fund diiding ecr s is mgniude s ie ŝ = s/ s There re hree uni ecrs i j nd k h re priculrl useful in ecr nlsis i j nd k re in he direcin f nd z es respeciel The re perpendiculr ech her nd herefre linerl independen ie if pi + qj + rk = hen p = q = r = k z j Emple An erplne is 5 km NE f Melurne Airpr nd i is fling n liude f km Resle is psiin ecr frm Melurne Airpr in i j nd k cmpnens The re respeciel pining es nrh nd up Wh is he srigh-line disnce f he plne frm he runw? = 5cs i OP 45 5 N P ( 5 cs45 5sin 45 ) sin 45 j + k = 5 i + 5 j + k Disnce frm he runw = OP = 5 + = 6 km Emple Find he mgniude f s = i j + 5k nd uni ecr in he direcin f s 5 s = s = + ( ) + 5 = Uni ecr in he direcin f s: 5 ŝ = s/ s = (i j + 5k)/ = ( i j + 5k) 5 i Sclr (d) prduc f w ecrs Sclr (d) prduc f w ecrs is defined mee he requiremens f mn phsicl siuins An ecr cn e wrien in erms f i j nd k eg s = i j + 5k s 5k i s i j nd k re linerl dependen j eg in phsics wrk dne W jules frce F newns in displcing n jec s meres is F W = Fs csθ θ s If he sclr prduc f F nd s is defined s Fs = Fs csθ hen W = Fs The sclr prduc f w ecrs gies sclr Ne: () In sclr prducθ is he ngle eween w ecrs h re plced il il () rr = r r = rr Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus

3 Emple Find he sclr prduc f he w ecrs 5 5 Emple Shw h i + j + 4k is perpendiculr i + j k (i + j + 4k)(i + j k) = = he w ecrs re prllel Plce he w ecrs il il θ = 6 Sclr prduc = 5 5cs 6 = 5 6 Emple Find c such h i + j k is perpendiculr i + cj + 6k (i + j k)(i + cj + 6k) = + c 6 = c = Prllel nd perpendiculr ecrs Tw ecrs p nd q re prllel if ne equls he her muliplied cnsn p = αq Tw prllel ecrs re linerl dependen The ngle eween w prllel ecrs is eiher zer r 8 Fs = Fs r Fs Tw ecrs re perpendiculr ( θ = 9 ) if Fs = Cnersel Fs = if w ecrs re perpendiculr Tw perpendiculr ecrs re linerl independen Ne h ii = jj = kk = nd ij = jk = ki = Sclr prduc f ecrs in i j k cmpnens θ If F nd s re in erms f i j nd k ie F = i + j + ck nd s = i + j + zk hen Fs = ( i + j + ck )( i + j + zk ) = + + cz Since Fs = Fs csθ F s + + cz cs θ = Fs + + cz Hence csθ = + + c + + z Emple Shw h p = i + j + 4k is prllel q = i + j + k 4 6 q = 4 i + 6 j + k = i + j + 4 k = (i + j + 4k) = p p nd q re prllel Emple 4 Find ecr perpendiculr g = i j + k Le r = li + mj + nk e ecr perpendiculr g rg = l m + n = Le m = nd n = l = Hence r = i + k is ecr (u f n infinie numer f hem) perpendiculr g Emple 5 Find uni ecr perpendiculr p = i j nd () n he sme plne () n n he sme plne Le u = li + mj + nk e uni ecr p = i j l + m + n = l + m + n = nd l m = () If u is n he sme plne s p hen he k cmpnen f u is zer ie n = Hence l + m = () nd l m = () Sle he simulneus equins fr l nd m: Frm () m = l () Susiue () in () 4 l + l = l = l = ± = ± m = ± 9 9 Hence u = i + j r i j () u (r u) is uni ecr p u n n he sme plne s p I cnins i j nd k cmpnens l + m + n = () nd l m = () Frm () m = l () Susiue () in () 4 l + l + n = l + n = n = ± l Chse l = = hen m = = nd 9 n = ± l 9 = ± Hence u = i + 9 u = i j = ± j + k k r Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus

4 Direcin csines Cnsider ecr h = i + j + ck nd he uni ecr in he direcin f h is: c ĥ = i + j + k + + c + + c + + c where + + c = h Le he ngles h he uni ecr mke wih he nd z- es e α β γ respeciel z c Emple Find he ngles h s = i j k mkes wih he es s = 6 ŝ = (i j k) = i j k cs α = cs β = nd csγ = Hence α 66 β 45 nd γ 4 Emple 4 nd c re hree rhgnl (ie muull perpendiculr) ecrs Find n epressin fr he csine f he ngle eween ( + + c) nd c Cnsider he sclr prduc f w ecrs: pq ( + + c)c = + + c c cs θ = pq csθ h c + c + cc = ( + + c)( + + c) c csθ γ β α Since nd c re rhgnl = c = nd c = Hence c = + + c c csθ nd csθ = c + + c csα = + + c cs β = nd csγ = c + + c + + c Sclr nd ecr reslues Insed f resling ecr in i j nd k cmpnens i cn e resled in her ws eg ecr q cn e resled in w perpendiculr cmpnens ne cmpnen q prllel nher ecr s nd he her cmpnen q s Hence ĥ = csα i + cs β j + cs γ k cs αcs β csγ re clled he direcin csines f h q q q s Emple Vecr r hs mgniude f nd i mkes ngles f 45 nd 6 respeciel wih i j nd k Epress r in erms f i j nd k r = cs i + cs 45 j + cs 6 k r = 5 i + 5 j + 5 k Emple Find he mgniude nd direcin csines f i 4j + 5k Mgniude = + ( 4) + 5 = Uni ecr = i j + k Direcin csines: cs α = = cs β = = nd cs γ = = q nd q re respeciel clled he sclr nd ecr reslues f q prllel s (r he sclr nd ecr prjecins f q n s) q nd q re respeciel clled he sclr nd ecr reslues f q perpendiculr s q = qŝ nd q = q ŝ = (qŝ)ŝ q = q q nd q = q q Emple Find he prjecin f 4i + k n i + j k ie sclr reslue f 4i + k in he direcin f i + j k Le h = 4i + k nd g = i + j k Uni ecr in he direcin f g: ĝ = ( i + j k) 4 The prjecin f h n g: hĝ = (4i + k) ( i + j k) = ( ) = Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus 4

5 Emple Find he sclr nd ecr reslues f = i j + k in he direcin f = i j nd perpendiculr â = (i j) 8 Sclr reslue f prllel : â = ( + 6) = Vecr reslue f prllel : 8 8 (â)â = (i j) = (i j) Vecr reslue f perpendiculr : 8 (â)â = (i j + k) (i j) = i j + k Sclr reslue f perpendiculr : 8 (â)â = + + = Emple Use ecrs pre Phgrs herem nd he csine rule B c c B θ C A C A Inrduce ecrs nd c s shwn e simplif he nins Phgrs herem: c = cc = ( )( ) cc = + Since = = c = + The csine rule: c = cc = ( )( ) cc = + Since = = cs θ = csc c = + csc Emple Pre h he ngle suended dimeer in circle is righ ngle A C Vecr prfs f simple gemeric resuls Emple Pre h he dignls f rhmus re perpendiculr Cnsider rhmus OPQR OP = PQ = QR = OR Inrduce ecrs p nd q frm cenre O B nd C respeciel O q p B O R P Q AO = p AC = AO + OC = p + q CB = OB OC = p q AC CB = (p + q)(p q) = pp qq = p q = ecuse p = q = rdius f he circle Hence AC CB ie ACB = 9 Emple 4 Pin P diides AB ccrding he ri m : n Find ecr OP in erms f m n nd OQ = OP + OR RP = OP OR OQ RP = ( OP + OR) ( OP OR) = OP OP OP OR + OR OP OR OR = OP OR = Since OQ nd RP re nn-zer ecrs nd OQ RP = he dignls re perpendiculr A m P n B O m OP = OA + AP = OA + AB m + n m = + m + n ( ) = (m + n + m m) m + n = (n + m) m + n Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus 5

6 Emple 5 Pre h he medins f ringle re cncurren nd he inersecin risecs ech medin A P O N AM nd BN re medins nd CP is line segmen pssing hrugh he inersecin O f AM nd BN Le OA = OB = nd OC = c OM = ( + c) nd ON = ( + c) Le m n nd p e sme psiie cnsns such h OM = m ON = n nd OP = p c m = ( + c) nd n B c = ( + c) Frm he ls w equins m = n hence ( m) ( n) = Since nd re n prllel he re independen nd hence m = nd ( n) = ie m = n = = ( + c) = c AP = AO + OP = pc nd AB = = c Le AP = k AB where k is psiie cnsn pc = k( c) ie (k ) + (k p)c = Since nd c re independen k = nd k p = ie k = nd p = P is he mid-pin f AB nd CP is medin Hence he hree medins re cncurren O nd he inersecin O risecs ech medin Vecr equins prmeric equins nd cresin equins (-dimensinl) As pricle mes is psiin chnges wih ime Is psiin cn e descried wih psiin ecr r() h is epressed in erms f i nd j cmpnens r() = ()i + ()j This equin is clled ecr equin nd he prmeer = () nd = () re w recngulr cmpnens in erms f he prmeer The re clled prmeric equins M C Emple Skech he lcus f pricle wih is psiin descried r() = i ( )j Vecr equin: r() = i ( )j Prmeric equins: = = ( ) Cresin equin: Frm he secnd equin = susiue in he firs equin = ( ) = ± + Since = ( ) = + Emple Skech he lcus f pricle wih psiin ecr r() = ( sin + ) i + ( cs ) j nd Vecr equin: r() = ( sin + ) i + ( cs ) j nd Prmeric equins: = sin + sin = + = cs cs = Cresin equin: + Since sin + cs = + = ie ( ) + ( ) = The lcus is n ellipse cenred ( ) A = he pricle π is = = A = i is = 5 = 6 nd mes clckwise he pricle srs ( ) When he prmeer is elimined frm he w prmeric equins we in cresin equin f he lcus (ph) f() = Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus 6

7 Emple Skech he lcus f pricle wih psiin gien r() = + i + j > Vecr equin: r() = + i + j > Prmeric equins: = + () = () Cresin equin: + () + () + = = + Susiue in () = + + ( + ) + 4 = ( + ) = ( + ) 4 ( + ) + + = = 4 ie = 4 4 The lcus f he pricle is hperl Fr > > he lcus cnsiss f he righ hnd rnch nl A + he pricle mes upwrds lng he hperl Emple 4 A pricle hs psiin ecr r() = + i + j > Find he cresin equin nd skech he grph f is lcus Cmpre he min f his pricle wih h in emple Vecr equin: r() = + i + j > Prmeric equins: = + () = () Cresin equin: () + () + + = = + Susiue in () = + = The pricle hs he sme lcus u differen speed Emple 5 Tw ships re n cllisin curse Their psiin ecrs re r() = ( + )i + (7 +6)j nd r() = (5 + 7)i + (4 + )j where When nd where d he ships cllide? The w ships cllide when he re he sme plce nd he sme ime Le ( + )i + (7 +6)j = (5 + 7)i + (4 + )j Eque crrespnding cmpnens: 4 + = nd 7 +6 = 4 + = A = he ships re he sme psiin r() = i + j Emple 6 The psiin ecrs fr pricles A nd B re r A () = ( + ) i + ( + ) j nd r B () = ( + 5 4) i +j respeciel Find n epressin fr heir seprin ime When nd where d he cllide? r A r B r A r B r B r A ( = + 5 4) i +j [ ( + ) i + ( + ) j] = ( 4 + ) i + ( ) j Seprin = r B r A = ( 4 + ) + ( ) The pricles cllide when heir seprin is zer ie ( 4 + ) + ( ) = ( 4 + ) + ( ) = ( ) ( ) + ( ) = ( ) ( ) + ) = = nd hence r = i + 9j is where he cllide Differeniin f ecr wih respec ime Gien ecr r() = ()i + ()j is firs nd secnd deriies wih respec re respeciel d d d d d r = i + j nd d d d r = d d d i + j d Deriies f psiin nd elci ecrs If r() is psiin ecr is firs deriie is elci ecr () nd he secnd deriie is n ccelerin ecr () d d d () = r () = r () = d d r d Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus 7

8 Emple The psiin ecr f n jec ming in plne is gien r() = i + j () Find is elci speed nd ccelerin when = () Drw he grph f he lcus nd he elci nd ccelerin ecrs = () r() = i + j 4 ( ) + ( ) = 9 4 = + d () = r = i + j d d () = = 6 i + j d A = = i + j = = 6i + j () = = = = Since 4 7 Emple Find he elci ccelerin nd speed f pricle whse psiin ecr is gien r() = e i + e j Skech he ph f he pricle nd drw he elci nd ccelerin ecrs = r() = e i + e j () d = r d = e i ( ) e + ( e ) = e + e e j d = () = = e i + e j d A = = i j = i + j Prmeric equins: = e = e Cresin equin: = e = = When = = e nd = he pricle srs frm ( ) Emple A pricle mes s h is psiin ecr ime is gien r = cs()i + sin()j fr () Find he elci nd ccelerin ime () Shw h in his cse is perpendiculr r nd is ppsie in direcin r (c) Skech he lcus f he pricle nd shw he elci nd ccelerin ecrs ime () r = cs()i +sin()j = d r = 6 sin( ) i + 6 cs() j d d = = cs() i sin() j d () r = 8 sin( ) cs() + 8 sin() cs() = r = cs() i sin() j = 4( cs() i + sin() j) = 4 r Hence is ppsie in direcin r (c) = cs() = cs() = sin() = sin( ) Use sin A + cs A = elimine : + = + = 9 The lcus is circle f rdius nd cenre ( ) A = = cs = = sin = A = > nd > nd mes niclckwise The pricle srs frm ( ) Emple 4 An jec mes in ph wih psiin ecr r() = cs i + sin j () Shw h he ph is ellipicl () Shw h he ccelerin pins wrds he rigin () = cs = sin = sin Elimine + = Ph is ellipicl 4 () r() = cs i + sin j = sin i + cs j = cs i sin j = (cs i + sin j) = r is ppsie r nd hence i is wrds he rigin Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus 8

9 Anidiffereniin f ecr wih respec ime Gien ccelerin () = p()i+ q()j elci is () = () d = () psiin is r() = () d p i d + q() j d nd Emple Gien he ccelerin = cs i + sin j nd when = r = i nd = j f pricle () Shw h he speed f he pricle is cnsn () Shw h he elci nd he ccelerin re lws perpendiculr (c) Shw h he ccelerin is lws wrds he cenre f he circulr ph (d) Shw h he ph is circulr (e) Skech he ph nd shw he direcin f min f he pricle ime () = cs i + sin j = cs i d + sin j d = sin i cs j + c where c is cnsn ecr When = = j c = = sin i cs j Hence = ( sin ) + ( cs) = 4 = cnsn Emple Gien he ccelerin = j nd when = he psiin is r = nd he elci is = 7i + j () Find he psiin ecr ime () Descrie he ph (c) A wh ime is he -crdine he sme s h =? (d) Wh is he -crdine h ime? () = j = j d = j + c c is cnsn ecr When = he elci is = 7i + j c = 7i + j Hence = j + 7i + j = 7i + ( ) j r = 7 i d + ( ) j d = 7 i + ( 5 ) j + d where d is cnsn ecr When = he psiin r = d = Hence r = 7 i + ( 5 ) j () = 7 = 5 5 Elimine = 5 = = ( 4) = ( ) + 5 = ( 7) Ph is prl ere (75) 5 () = 4 sin cs 4cs sin = is (c) r = sin i d cs j d = cs i sin j + d where d is cnsn ecr When = r = i d = r = cs i sin j = r Hence is lws wrds he rigin ie he cenre f he circulr ph (d) = cs = cs = sin = sin Use sin A + cs A = elimine : + = + = 4 The ph is circle f rdius nd cenre ( ) (e) r 7 4 (c) A = = When = 5 = = (d) A = = = 4 Emple A pricle is prjeced he rigin wih elci gien = i + j + k nd i hs cnsn ccelerin = k The uni ecr k is pining ericll upwrd () Find he elci ecr f he pricle ime () Find he psiin ecr f he pricle ime (c) Find he mimum heigh e he rigin (d) Find he disnce f he pricle frm he rigin when i reurns he sme leel s he rigin () = k = k d = k + c where c is cnsn ecr When = he elci is = i + j +k c = i + j + k Hence = k + i + j + k = i + j + ( ) k () r = i d + j d + ( ) k d = i + j + ( 5 ) k + d where d is cnsn ecr When = he psiin r = d = Hence r = i + j + ( 5 ) k Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus 9

10 (c) A he highes pin he pricle mes hriznll ie he k cmpnen f is zer = = A = he k cmpnen f r is 5 = 5 = 5 he mimum heigh is 5 (d) When he pricle reurns he sring leel he k cmpnen f r is zer 5 = = A = r = i + j + ( 5 ) k = i + j disnce frm rigin = r() = + = The ls w emples re prjecile mins under cnsn ccelerin due gri g Elimining frm he ls w equins: = ( csθ ) = Susiue in csθ = ( ) g = ( ) g cs cs θ θ g = nθ This is he equin f he ( csθ ) lcus f he prjecile sisfing he cndiins sed in he pening senence A generl cnsiderin f prjecile min under cnsn ccelerin due gri ( D) Le = gj nd he iniil psiin nd elci re respeciel r = nd = csθ i + j Iniil speed θ csθ = gj = g j d = g j + c where c is cnsn ecr When = he elci is = csθ i + j c = csθ i + j Hence = g j + csθ i + j = csθ i + ( g) j r = csθ i d + ( g) j d = ( csθ ) i + (( ) g ) j + d where d is cnsn ecr When = he psiin r = d = r = ( csθ ) i + (( ) g ) j Hence ime he hriznl nd ericl cmpnens f he elci ecr re = csθ nd = g respeciel Als ime he hriznl nd ericl cmpnens f he psiin ecr re = ( csθ ) nd = ( ) g respeciel j i Rnge The mimum heigh is reched when he j cmpnen f is zer ie g = = g A = g ( ) he j cmpnen f r is = ( ) g = g ( ) mimum heigh = g Als mimum heigh is reched mid-rnge where csθ sin θ = ( csθ ) = ( csθ ) = = g g g he rnge f he prjecile is sin θ g Emple A pricle is prjeced wih speed f ms - 6 wih he hriznl frm he rigin Tke g = ms - () Find he equin f he ph f he pricle () Find he mimum heigh reched (c) Find he rnge f he pricle () = n 6 (cs 6 ) = ( ) (sin 6 ) () Mimum heigh = = = 5 m g sin θ sin (c) Rnge = = = m g Cprigh iuecm 6 Free dwnld & prin frm wwwiuecm D n reprduce her mens Vecrs nd ecr clculus