Vectors. Chapter. Introduction of Vector. Types of Vector. Vectors 1

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1 Vect 1 Chapte 0 Vect Intductin f Vect Phical quantitie haing magnitude, diectin and being la f ect algeba ae called ect. Eample : Diplacement, elcit, acceleatin, mmentum, fce, impule, eight, thut, tque, angula mmentum, angula elcit etc. If a phical quantit ha magnitude and diectin bth, then it de nt ala impl that it i a ect. F it t be a ect the thid cnditin f being la f ect algeba ha t be atified. Eample : The phical quantit cuent ha bth magnitude and diectin but i till a cala a it dibe the la f ect algeba. Tpe f Vect (1) Equal ect : T ect and ae aid t be equal hen the hae equal magnitude and ame diectin. () Paallel ect : T ect and ae aid t be paallel hen (i) th hae ame diectin. (ii) ne ect i cala (pitie) nn-e multiple f anthe ect. (3) nti-paallel ect : T ect and ae aid t be anti-paallel hen (i) th hae ppite diectin. (ii) ne ect i cala nn-e negatie multiple f anthe ect. (4) Cllinea ect : When the ect unde cnideatin can hae the ame uppt hae a cmmn uppt then the cnideed ect ae cllinea. (5) Ze ect ( 0) : ect haing e magnitude and abita diectin (nt knn t u) i a e ect. (6) Unit ect : ect diided b it magnitude i a unit ect. Unit ect f i  (ead a cap hat). Since, ˆ ˆ. Thu, e can a that unit ect gie u the diectin. (7) thgnal unit ect ˆ i, ˆ j and kˆ ae called thgnal unit ect. Thee ect mut fm a ight Handed Tiad (It i a cdinate tem uch that hen e Cul the finge f ight hand fm t then e mut get the diectin f alng thumb). The ˆ i, ˆ j, kˆ ˆi, ˆj, kˆ (8) Pla ect : Thee hae tating pint pint f applicatin. Eample diplacement and fce etc. (9) ial Vect : Thee epeent tatinal effect and ae ala alng the ai f tatin in accdance ith ight hand ce ule. ngula elcit, tque and angula mmentum, etc., ae eample f phical quantitie f thi tpe. nticlck ie tatin ial ect i f tatin Fig. 0. Clck ie tatin i f tatin ial ect (10) Cplana ect : Thee ( me) ect ae called cplana ect if the lie in the ame plane. T (fee) ect ae ala cplana. kˆ ĵ î Fig. 0.1

2 Vect Tiangle La f Vect dditin f T Vect If t nn e ect ae epeented b the t ide f a tiangle taken in ame de then the eultant i gien b the cling ide f tiangle in ppite de. i.e. (1) Magnitude f eultant ect In N, N c N c N in N in In N, e hae and, then N c Fig. 0.4 ( c) ( in ) N c c in (c in ) c c c () Diectin f eultant ect : If i angle beteen c If make an angle ith, then in N, tan N N N N in tan c Paallelgam La f Vect dditin If t nn e ect ae epeented b the t adjacent ide f a paallelgam then the eultant i gien b the diagnal f the paallelgam paing thugh the pint f inteectin f the t ect. (1) Magnitude Since, N CN ( N) CN c N Fig. 0.3 in Special cae : hen = 0 hen = 180 hen = 90 () Diectin tan CN N in c Plgn La f Vect dditin If a numbe f nn e ect ae epeented b the (n 1) ide f an n-ided plgn then the eultant i gien b the cling ide the n th ide f the plgn taken in ppite de. S, e. C D E C CD DE E Fig. 0.6 Nte : eultant f t unequal ect can nt be eultant f thee c-plana ect ma ma nt be e eultant f thee nn c- plana ect can nt be e. Subtactin f ect Since, ( ) and c c(180 ) Since, E c(180 E ) c D Fig. 0.5 D c C c C C N in c

3 Vect 3 um in tan 1 c in(180 ) and tan c(180 ) ut in( 180 ) in and c( 180 ) c in tan c elutin f Vect Int Cmpnent diff Fig. 0.7 If make an angle ith ai, ith ai and ith ai, then c c c l m n Cnide a ect in X-Y plane a hn in fig. If e da thgnal ect and alng and ae epectiel, b la f ect additin, N a f an ect ˆi and ˆj nˆ, ˆi ˆj (i) ut fm figue and c (ii) in (iii) Since and ae uuall knn, Equatin (ii) and (iii) gie the magnitude f the cmpnent f alng and -ae epectiel. Hee it i th t nte nce a ect i eled int it cmpnent, the cmpnent themele can be ued t pecif the ect a (1) The magnitude f the ect i btained b quaing and adding equatin (ii) and (iii), i.e. () The diectin f the ect i btained b diiding equatin (iii) b (ii), i.e. tan ( / ) tan 1 ( / ) ectangula Cmpnent f 3-D Vect Y q ˆi ˆj kˆ Y Fig. 0.8 X Whee l, m, n ae called Diectin Cine f the ect and l m n c c c 1 Nte : When a pint P hae cdinate (,, ) then it pitin ect P i ˆ j ˆ kˆ When a paticle me fm pint ( 1, 1, 1 ) t (,, ) then it diplacement ect ( )ˆi ( )ˆj ( ) kˆ Scala Pduct f T Vect (1) Definitin : The cala pduct ( dt pduct) f t ect i defined a the pduct f the magnitude f t ect ith cine f angle beteen them. Thu if thee ae t ect and haing angle beteen them, then thei cala pduct itten a. c () Ppetie : (i) It i ala a cala hich i pitie if angle beteen the ect i acute (i.e., < 90 ) and negatie if angle beteen them i btue (i.e. 90 < < 180 ). (ii) It i cmmutatie, i.e... (iii) It i ditibutie, i.e..( C).. C (i) b definitin. c. i defined a Fig X Z Fig. 0.9

4 4 Vect. The angle beteen the ect c 1 () Scala pduct f t ect ill be maimum hen c ma 1, i.e. 0, i.e., ect ae paallel (. ) ma (i) Scala pduct f t ect ill be minimum hen c min 0, i.e. 90 (. ) min 0 i.e. if the cala pduct f t nne ect anihe the ect ae thgnal. (ii) The cala pduct f a ect b itelf i temed a elf dt pduct and i gien b ( ). c i.e.. (iii) In cae f unit ect nˆ n ˆ. nˆ 1 1 c0 1 n ˆ. nˆ ˆ i.ˆi ˆ j.ˆj k ˆ. kˆ 1 (i) Ptential eneg f a diple U : If an electic diple f mment p i ituated in an electic field E a magnetic diple f mment M in a field f inductin, the ptential eneg f the diple i gien b : U E p. E and U M. Vect Pduct f T Vect (1) Definitin : The ect pduct c pduct f t ect i defined a a ect haing a magnitude equal t the pduct f the magnitude f t ect ith the ine f angle beteen them, and diectin pependicula t the plane cntaining the t ect in accdance ith ight hand ce ule. itten a C Thu, if and ae t ect, then thei ect pduct i a ect C defined b C in nˆ (i) In cae f thgnal unit ect ˆ i. ˆj ˆj. kˆ kˆ.ˆi 11 c 90 0 ˆ i, ˆ j and ˆk, () In tem f cmpnent ( i j k ).( i j k ) ]. [ Z (3) Eample : (i) Wk W : In phic f cntant fce k i defined a, W Fc (i) ut b definitin f cala pduct f t ect, F. Fc (ii) S fm eq n (i) and (ii) W F. i.e. k i the cala pduct f fce ith diplacement. (ii) Pe P : W F. dw d F. [ F i cntant] P F. i.e., pe i the cala pduct f fce ith dw elcit. P and d (iii) Magnetic Flu : Magnetic flu thugh an aea i gien b d dc (i) ut b definitin f cala pduct. d d c...(ii) S fm eq n (i) and (ii) e hae d. d.d d Fig The diectin f, i.e. C i pependicula t the plane cntaining ect and and in the ene f adance f a ight handed ce tated fm (fit ect) t (ecnd ect) thugh the malle angle beteen them. Thu, if a ight handed ce he ai i pependicula t the plane famed b and i tated fm t thugh the malle angle beteen them, then the diectin f adancement f the ce gie the diectin f i.e. C () Ppetie (i) Vect pduct f an t ect i ala a ect pependicula t the plane cntaining thee t ect, i.e., thgnal t bth the ect and, thugh the ect and ma ma nt be thgnal. (ii) Vect pduct f t ect i nt cmmutatie, i.e., [but ] Fig. 0.1 Hee it i th t nte that in i.e. in cae f ect and magnitude ae equal but diectin ae ppite. (iii) The ect pduct i ditibutie hen the de f the ect i tictl maintained, i.e.

5 Vect 5 ( C) C (i) The ect pduct f t ect ill be maimum hen in ma 1, i.e., 90 [ ] nˆ ma i.e. ect pduct i maimum if the ect ae thgnal. () The ect pduct f t nn- e ect ill be minimum hen in minimum = 0, i.e., [ ] min 0 (i) Fce n a chaged paticle q ming ith elcit in a magnetic field i gien b F q( ) () Tque n a diple in a field p E and M Lami' Theem In an C ith ide a, b, c in in in a b c 180 E i.e. if the ect pduct f t nn-e ect anihe, the ect ae cllinea. (i) The elf c pduct, i.e., pduct f a ect b itelf anihe, i.e., i null ect in 0 nˆ 0 c b 180 (ii) In cae f unit ect nˆ nˆ 0 that ˆi ˆi ˆj ˆj kˆ kˆ 0 (iii) In cae f thgnal unit ect, ith ight hand ce ule : ĵ ˆ i, ˆ, j kˆ in accdance ĵ 180 Fig i.e. f an tiangle the ati f the ine f the angle cntaining the ide t the length f the ide i a cntant. a F a tiangle he thee ide ae in the ame de e etablih the Lami' theem in the flling manne. F the tiangle hn î kˆ î a b c 0 [ll thee ide ae taken in de] (i) a b c (ii) kˆ Fig ˆi ˆj kˆ, ˆj kˆ ˆi and kˆ ˆi ˆj nd a c pduct i nt cmmutatie, ˆj ˆi kˆ, kˆ ˆj ˆi and ˆi kˆ ˆj () In tem f cmpnent Pe-multipling bth ide b a a ( a b) a c 0 ab a c a b c a (iii) Pe-multipling bth ide f (ii) b b b ( a b) b c b a b b b c ˆi ˆj kˆ ˆ( i ) ˆ( j ) kˆ( ) (3) Eample : Since ect pduct f t ect i a ect, ect phical quantitie (paticulal epeenting tatinal effect) like tque, angula mmentum, elcit and fce n a ming chage in a magnetic field and can be epeed a the ect pduct f t ect. It i ell etablihed in phic that : (i) Tque F (ii) ngula mmentum L p (iii) Velcit a b b c a b b c (i) Fm (iii) and (i), e get a b b c c a Taking magnitude, e get a b b c c a ab in( 180 ) bcin( 180 ) cain( 180 ) abin bcin cain Diiding thugh ut b abc, e hae in in in a b c elatie Velcit (1) Intductin : When e cnide the mtin f a paticle, e aume a fied pint elatie t hich the gien paticle i in mtin. F eample, if e a that ate i fling ind i

6 6 Vect bling a pen i unning ith a peed, e mean that thee all ae elatie t the eath (hich e hae aumed t be fied). S X Fig N t find the elcit f a ming bject elatie t anthe ming bject, cnide a paticle P he pitin elatie t fame S i hile elatie t S i. If the pitin f fame S elatie t S at an time i then fm figue, SS Diffeentiating thi equatin ith epect t time d Y d ds S Y S ' S S P ' X S S ith e elatie t the cente f eath, the elcit f atellite elatie t the uface f eath e e S if the atellite me fm et t eat (in the diectin f tatin f eath n it ai) it elcit elatie t eath' uface ill be e e nd if the atellite me fm eat t et, i.e., ppite t the mtin f eath, e ( ) e (4) elatie elcit f ain : If ain i falling eticall ith a elcit and an bee i ming hintall ith peed M the elcit f ain elatie t bee ill be e M hich b la f ect additin ha magnitude M M M diectin tan 1 ( M / ) ith the etical a hn in fig. [a d / ] S S S S () Geneal Fmula : The elatie elcit f a paticle P 1 ming ith elcit 1 ith epect t anthe paticle P ming ith elcit i gien b, 1 then : = 1 (i) If bth the paticle ae ming in the ame diectin then : 1 1 (ii) If the t paticle ae ming in the ppite diectin, 1 1 (iii) If the t paticle ae ming in the mutuall pependicula diectin, then: P 1 1 Fig (i) If the angle beteen 1 / 1 1 c. 1 and be, then (3) elatie elcit f atellite : If a atellite i ming in equatial plane ith elcit P and a pint n the uface f eath (5) elatie elcit f imme : If a man can im elatie t ate ith elcit and ate i fling elatie t gund ith elcit elcit f man elatie t gund M ill be gien b: M M elcit M M M, i.e., M Fig M S if the imming i in the diectin f fl f ate, nd if the imming i ppite t the fl f ate, (6) Cing the ie : Suppe, the ie i fling ith. man can im in till ate ith elcit m. He i tanding n ne bank f the ie and ant t c the ie, t cae aie. (i) T c the ie e htet ditance : That i t c the ie taight, the man huld im making angle ith the upteam a hn. Upteam m Fig Dnteam

7 Vect 7 ect can hae an numbe, een infinite cmpnent. (minimum cmpnent) Hee i the tiangle f ect, in hich, m. Thei eultant i gien b. The diectin f imming make angle ith upteam. Fm the tiangle, e find, c l m m in m Whee i the angle made b the diectin f imming ith the htet ditance () ac the ie. Time taken t c the ie : If be the ih f the ie, then time taken t c the ie ill be gien b t1 m (ii) T c the ie in htet pible time : The man huld im pependicula t the bank. The time taken t c the ie ill be: t m Upteam Fig In thi cae, the man ill tuch the ppite bank at a ditance dn team. Thi ditance ill be gien b: t m ll phical quantitie haing diectin ae nt ect. F eample, the electic cuent pee diectin but it i a cala quantit becaue it can nt be added multiplied accding t the ule f ect algeba. ect can hae nl t ectangula cmpnent in plane and nl thee ectangula cmpnent in pace. m Dnteam Flling quantitie ae neithe ect n cala : elatie denit, denit, icit, fequenc, peue, te, tain, mdulu f elaticit, pin ati, mment f inetia, pecific heat, latent heat, ping cntant ludne, eitance, cnductance, eactance, impedance, pemittiit, dielectic cntant, pemeabilit, uceptibilit, efactie inde, fcal length, pe f len, ltman cntant, Stefan cntant, Ga cntant, Gaitatinal cntant, dbeg cntant, Planck cntant etc. Ditance ceed i a cala quantit. The diplacement i a ect quantit. Scala ae added, ubtacted diided algebaicall. Vect ae added and ubtacted gemeticall. Diiin f ect i nt alled a diectin cannt be diided. Unit ect gie the diectin f ect. Magnitude f unit ect i 1. Unit ect ha n unit. F eample, elcit f an bject i 5 m 1 due Eat. 1 i.e. 5m due eat. 1 5 (Eat) ˆ m Eat 1 5m S unit ect ˆ ha n unit a Eat i nt a phical quantit. Unit ect ha n dimenin. ˆi.ˆi ˆj.ˆj k ˆ. kˆ 1 ˆi ˆi ˆj ˆj kˆ kˆ 0 ˆi ˆj kˆ, ˆj kˆ ˆ, i kˆ ˆi ˆj ˆi.ˆj ˆj. kˆ kˆ.ˆi 0 0. l 0 ut ecaue and i cllinea ith Multiplicatin f a ect ith 1 eee it diectin. If, then = and ˆ ˆ. If 0, then = but ˆ ˆ. Minimum numbe f cllinea ect he eultant can be e i t. Minimum numbe f cplane ect he eultant i e i thee. Minimum numbe f nn cplane ect he eultant i e i fu. T ect ae pependicula t each the if. 0.

8 8 Vect T ect ae paallel t each the if 0. Diplacement, elcit, linea mmentum and fce ae pla ect. ngula elcit, angula acceleatin, tque and angula mmentum ae aial ect. Diiin ith a ect i nt defined becaue it i nt pible t diide ith a diectin. Ditance ceed i ala pitie quantit. The cmpnent f a ect can hae magnitude than that f the ect itelf. The ectangula cmpnent cannt hae magnitude geate than that f the ect itelf. When e multipl a ect ith 0 the pduct becme a null ect. The eultant f t ect f unequal magnitude can nee be a null ect. Thee ect nt ling in a plane can nee add up t gie a null ect. quantit haing magnitude and diectin i nt neceail a ect. F eample, time and electic cuent. Thee quantitie hae magnitude and diectin but the ae cala. Thi i becaue the d nt be the la f ect additin. phical quantit hich ha diffeent alue in diffeent diectin i called a ten. F eample : Mment f inetia ha diffeent alue in diffeent diectin. Hence mment f inetia i a ten. the eample f ten ae efactie inde, te, tain, denit etc. The magnitude f ectangula cmpnent f a ect i ala le than the magnitude f the ect If, then, and. If C. if C 0, then, and C lie in ne plane. If C, then C i pependicula t a ell a. If, then angle beteen and i 90. eultant f t ect ill be maimum hen = 0 i.e. ect ae paallel. ma P Q PQ c 0 P Q eultant f t ect ill be minimum hen = 180 i.e. ect ae anti-paallel. min P Q PQ c180 P Q Thu, minimum alue f the eultant f t ect i equal t the diffeence f thei magnitude. Thu, maimum alue f the eultant f t ect i equal t the um f thei magnitude. When the magnitude f t ect ae unequal, then min P Q 0 [ P Q ] Thu, t ect P and Q haing diffeent magnitude can nee be cmbined t gie e eultant. Fm hee, e cnclude that the minimum numbe f ect f unequal magnitude he eultant can be e i thee. n the the hand, the minimum numbe f ect f equal magnitude he eultant can be e i t. ngle beteen t ect and i gien b. c Pjectin f a ect in the diectin f ect. Pjectin f a ect in the diectin f ect. If ect, and C ae epeented b thee ide ab, bc and ca epectiel taken in a de, then C ab bc ca The ect ˆi ˆj kˆ i equall inclined t the cdinate ae at an angle f degee. If C, then. C 0. If. C 0, then. and C ae cplana. If angle beteen and i 45, then. If n 0 and n then the adjacent ect ae inclined t each the at angle / n. If C and C, then the angle beteen and i 90. l, and C can hae the flling alue. (i) = 3, = 4, C = 5 (ii) = 5, = 1, C = 13 (iii) = 8, = 15, C = 17.