F Chapter 10 KD OE _ ss dfumit_vectet - A Vectet navi? 3] equal _vectet :- vectors having 4. Lipe vectors :— (hoving sam s unbifea vectors *~ (vectors having " nto? ve 5): vectors havira, Same Line 7 en Vector Aaeb2 es |Deirition Me 1,4 vere. ¢ et, re Cote — veri . a velocity oceolatatior d both A quar ity thot at, wngnitucle and odeuchon | Roplthe vaio n of a vectof =o \* vecte can be sepApseriteot by a dLetecteo Lie Segment such at fie’ Length “of fe Lene Samet aw tHe magni tucle of 4ho vetlel. vectors ate andl, ldleno teal by tho names a, b, ete- AG o called magniticle Dehoreal by [at > — > P vect AB apes _of _veciets | geto_vectot=(0}) waanitucle one ts cables areteed amit vector. , 4 Sao A veces WwW +Po otifection a Bees Be und : limit vectet in. the olivection y 8 & AB ne Rae 8 eat Tae" | game mpagnitucls t Abection- 2. chiaection) oppe: ofitector) : achon parallel Line of action:t ae eet Re Re arte Ee re treet LSet arg, Ne ca AT acpi F: 7. Copforad vectors t- Vectors Lyena, on He Same plo B Petition vectet :- lt & a veclet obtained by déerpig * | etn to a given point. d | Position vecter 2, ee p= of , PV of point P= or ei Neao tive of a_vecte? -a) ay Taiorrgge Low of Vectet_sobbition | ee \ 4a A > FQ veetots topzsents adljacerit sicker P 2 Biaryle Ankon mM the Same ereles (ésection),: thon Ho Sumi. Ropsasented by the thild sick taken’ i Aevesse otoles, Uo Blah AC aT Weak Prasalllelogiarcn of Veclet Adblition Final Hee diagonal ve cH.along Heath diagonal. = Seiler 8 ” Seala3 Mul tipLication ’ 5 Gate —_ ae ay oo o aruy vectey £ Swe be scalat, then, —11.a t @ vet cohose” vm nituale | . m7 Hmet that ef a? amd having ‘seomenaee io yma CE oY having, otha divedion of Q veces Cbg ate patallel ot collineat ¢ one vector os the scalod roebtipe ¥ the otha. | &) B= A Hes VD” @a) txpIwwrsion a Nector Using. position -geuetes AB = 6B -0F- |: = ov (8)- pv)ot Fetal re at et. fio He potition vectt of a ph R hich & ne ’ ™M the xoto won. m— Yd Beales taual (m=) [Midpoint Ferra] OR’ = pero Soren. 2 Comnponent form of a Vectet oP = at + ys +2 4 : loPl~ Jor aye? Scalar Component oe hos atieiesacihaitaiaedaiegae rad 2) YZ Ve ches Component = 24, 38, 2f P(2.4.2)he le two vectsss at puiadhl, thon +heil Scales Crqpone ore plopeitional. pest fod pont The vecte? joining ¢ the points P(z,.%, 2) to (OCS yay ao given by if PO = (-1) TF (4-¥)5 4 (Z,-2) 2) pen O(n. 25) Pa= 58-88 -Direction Cosine’ ¢ Ovtection Ratios Doection Angles ect el } The am rade by the vect with patrve 24,2 lamers ade Cnbled olitection anges . Denote? by of, B, t. ioc ’s(Disection Cosinot) Cosine of the lifecHon angles are cabled OCs. Denoresdts by 854, > = Ctod ot! fos B, os > Direction Ratios : Any 3 no's which aie. propeltional +o dlitection cesives are Calleol olitection Sates. <*> ORs =< kL KM, kKn> OP = xt+yt+22 , thon ORs= OG sae es! ,— > Vx24yth2? Vrtytz? Jrit ye Note ch Dd} OC's of M-auwk —»< 1,0,0> | c's ef y-axk—> . <a a 2) Lirmrtne = 41 ‘ A f _— w a unit vec 4:5) Given, thot, =1+ af nL, A b= 2743 Ia = = Fi ae = Vag s VE v) Pe etd: snot equal , because the conrespor chive scales Components a%0 ot equal. 2 He amit veedot NM Jo) ! tq°6 Given tht, a=wen that, B= ahsag-sk = atege3t aw = 474 35-22-(27) VO |= J+%43%2? = vag a —> Sy) Given ot, B= 444-28 Ns v s a & w cs ' fe *«S Mec s 4 5S aye oS Pee cls Dee =< ee i> == 1) Grwew, Plas.,2,) = 2, 3,0) ancl tia ain 3-4) 78 = fu-3)4 + Gs wtae-ny # = * =4 2 = Biz STA pee ee tz _10-¢ 10-2 3 wl ai. State +S ; ela JPR? = eo e pois : feast : = n-3-6 Ax@ ls0)' ae e ce Lee ae Sesprak |6) Giivent initial point (4, », @) Terminal p+ (ry¥,) =€5, 7) “Rea, veel = G1) 7 4 bHymy) = -#i+6f “Scales lemponets = ~7, 6 Nec tet Components = RFs) 6 SS 6) riven, B=1-IT , w= aes, Gea -6f-72 PiVre= oe ro eae = oh -49-k = 43k i Hore, a= 4F42k Ve | = Ji? = Ve- . ees ce oe a7 | z vé ve jo wea 8) Given, P(x,,4,,2,) =(1,2,3) @(x,,42,2) =4,5 6) FO = G-y4sea)9+@- VE Sahat +3'k \pGl = 3743743? = Jag = 3v5 > ileoeg al eas ert ee I ae : + A poe a ee Aralfind umit véctet Wes got Leu to pe Fine the unit veciy ¢ Dut (t) Syrn bef. Een) riven Hot, OP = 3P-2E amd SP= AtD’ intemal, (Aatio <#2 (mo m)=9:1) Oe = TOT s+ nlor min 2 90( 8748) Hy (387-207) = 20428432827 = can ’ a Zs ii) éztanall, (Lato 2 y . BR = 7 O@- OP A ORe OS.T 1 0 : A da MEN i eee = 3@¥)- Gra) a2 on 207+ B30 4907 ; ' * = -O+40" a2 \ABl= Vieatre? = \Bcl _* Garris a {Ac}? = 1242745? = 35° nc)? (BC) = 1AB)* 1 AABC _O Aigh-orrgfeal Triangle - fx 10a Cate ,pty pied en ee pO Ce ae ee. Ue oe as Ala 2)=Cary amt 8(%,%,2a)= Fa Y) 1 AB = Cita C2-ay + fl4ye = -ateaya? Mies 4 et ty pe - = 5 4 Yas fs pay One ots pe aw . Lewy Gwen let @=43548 DRee fay i = 10°} | Please, where 8 w He anga betow, a amo b? | o.b° & a Scolas qontity . a ) IE Peg Bake two nev- -3hto vectetk , Hen ¢ efile “1a iI) oh Diststntive) a. t= eS w Joduct in Component form a.%+a,4+a,k ond 4 bat +32 Regge(| then, Bh? = a,b, + Aghyt Ag b3 \ arafe Yur 4W0 Vec tts iv Component otra oe cea Qh) +; by a tata aa a,7+a24a; Corea SrA Notes 4 ees POW Bee As ' bo b3 s oly == yb, 4 Ay by +04 by = (seen ¢ Peduct of dascalas ran equal +e gate) Given thot, Brees? PP He ae ab wats, 1 Be ces ee | Sa 20) ta] ca b= ex! A) 6= 60 7% UE mA , av, ett Leer Ie anoe sie is ‘ eae that, am st- cardi een — tases Paes et sat v-b= ot eee tad (P4B).@-B)= 2492/6 0 =0 . @+3?) and, (@-) are A fat. J Given, 89) Bi=a les, o.b~ JR-P)* = 1ai2al/?- 24.8) = ++9-8 Pees ia ae «ke Cy iecluliad | eves, esha ue ndt copanet be. cup) & @ Unit tenvectss a) \el=4 @-2). ez = Be es gy Br > \el-1s8 => wl?=44 ‘ ‘yxl- 03‘ ‘eal “4 and T= 96. 99:n see alamt a7 -T- 2743, and B= sfe-2p HF Be tat Gy) +azb, +056 a tes”! 3+4 43 Vaiia,240,2 SHE? Vit oie cen mi fh) 0 af) E *: = 06H), 5 tt P-SAasiict) 3 72 Lt (ot eSya8) ze Po eb (ct+ tae) yal l> Joey s% 0 —_ PE = L(atesf rch). + 34- roe) eg a a7 Ol2 Ch ala PD - lox.B— asl’ > HP Psa CR +e. P35 fel? —— 5 ative, \a’| = \v"l 8 = 60° a7.y? = a 1 =a?!” cos 60" \ rt PV = 1210) cso | a Given, -a). (#407 x12 es on @ unit vected "" Pl= om a9) ete aa lars 21z1?= Aoct ; ect af Iz] oil ee Tah i ilew). (PB) lel tl? ee ae we lreree)' lta ralel faa were ' (a Gee diets ak “owt baa Teafet , j gr eg re | given, ab Pa | |(@+ab)= Ries Balas Girates) : as: Fatt yr ae tah sarpeae ey Gate Gra + + GWE Frise tot, @ +B) Te 2 (PHP) = 0 : b« = +Qta9)s + Graph] bard] =o — D&-32) tRtaa)t+ 0 = 0 | > %-a-0 >r=8 —_ = fi) given , #l=|P)=)e] =4 ae erea > sak ni ease Seaseiciey elas Welt s ‘aed Fa eee ep OSL ES “Fi { ww) IP +P+?]7= 0 Briss aerev aie nea S2f4] aH= -24 S eo oe, Il=5,fel=4 jieles #00 given, me ee)ne ae Dera ag B.@rre 3 Ee + peo a0) 2. (PHP) =0 SVCr +P.F=0—B O+O+@ |] wvsrav.erw.e=-o | fares [P= ares (Ps 2, afar? PER pee ard = 32% 4245 % 426 = Vtlét+as jorvrel® so : rPre |= VSO = Sar Ex lo-3 oN a= lelv sibel bra tele, Pa (awswile) lewis) = ettete- Brel eT? tePa) a Abo given, a. b= 0 i v Com Wave omy value. t \alGiven, F W=0 RK B=0 , a.4"=0 | Corser Se eo 5 = ROE Oe : * Pw =o , ton =O W BW =O. Thomempics -o-4, P-) Wa) [WEI bet 47.3’=0 2. This Conved3e & Pot Tew —, f= 1742°4,2 = 4) [82\2= a2¥ PP Hi%©: [Giver : yel=a | \aal= a alle | = ila : | | ' + ea MeWlcam 4 (for emit vectol) 34,168 i ie €q'14) Giver thot 5 oe Oa Qtr3} tak be Ttayrk " | : epeiesbien f aonb = A+6tQ 10 VF +274)? Zal) Giver thot ACateanss®) , aCMHI+3E) ond cGzi-£)But Bow o amen ft: PeAG ¢ BC nods ® Sonqe Lie. 6f acHen - yo 4,8,C awe ceblingal » ~~ Poi Ex 10-3 4 a) (44% | jech 4 4,4 iS ty Ofer my tim aon tf ERG a aaliaie as ojection of vette N439 47K on vectt FIG ae = dhe3ttat). GISred \99 -J' +02] = 702340 to Fe “Vinaee ‘ean taht” Subs Fay by 4 by Va?raztae Wiest Je ay B= wy (stare ' -) wt): e ae A Wen, OA=4423432 ,o8= a4 3432, O0’= 314 107)- a= O8-OF = 4449-47 R= &- 78 = See &) A= 80 Kut Be Sut 8 & a common pt. AG t BC nave. Some Dine ¥ actor “Goins A,B, C ate colli A, B, C oie coinaas. Bow Prockuct Vectss Phooluct) prodluct eo VP ER db Lemotet by Ey one be = IE] sine A , wowse, A ty Ha unit vector 1 to both, wer.Neve A] ax b? & a vectet quantita 2 Diwerion of dnheneh, a Wer, Her we a ~ of wee sopsasent Liagon of Avo £7 = tale.P| af 99 | : ye (3,1) : i : VEN 8 hin 12,3) iven, O£= tagak i re iR =4,A 35) Gwen, @=394 9440 Va Tjek \ae. AC) = vas ge = Vai thee of FRAAGC = (i eal - \ &_l0-+ ee Ginen thot d= "%y Bs my Wg C1) CO l= WS 7/3 ==e%> Cos = YQ P¥=5 77, ~ cs the eso ve @) inte, me ky [Me ts 6 Row hat Rm? in? =|a) tos O~ S$ & O ce ackhel | oa “8 =60= */3 4 einfa nt oe Meee Also we Pe Li tmp tnk @ Bn 2. Required aan 2 w= fittmid ne t@: a as FE os Be a ae Scalas com ponent +, +, wm ns “Ao vectH Compone 14 ee) aDwa Tape and Toe Wwe) 3 ey ee " 4 () e961) Hy lagtiat ‘ aera Vta2(ia) Pf some nie? oy bath Plata" eX 7 "Se Faget WONG 26 i - neh Cy 1 aa) 4G : VV 490440 (=a s ( The verter Peat. 4 & Bad " Wwe, 2* pk n= -? fads KB = (8 -dk= 2% AP - 8B-0R= -3 a a e ‘ed "Re x ABS : : ss oe il 0 peo 36) 4tC3) Aad ao - eee) ; | Q = 2 wits “option (6) —=——= —- Oe eee en Se Given thot, le] => iy) = » led] =4 sin = !Pxd"] oak 7 ie) * Raepine, Ry Lo <> Pree 6 L apeove oor. Bresfen i 9 6-U-1-a =O ae > 5-0 =0 | a= Sebstshaing @ im O£O, we get Bert fi 3t- imeoeyoS eae ‘ q B). (Pep) = pete wwe ww eI! ! =trlt+a0.P trl? = yell? Perry .o vP=o] Merce et ve Gos « Ale qiven te et 2 ase meetin Oy pe J QeV=V.C= 2.6 =0 xt 9, , 8, , 0, be he angle made by the vectos (rv+e) with 7, 2 sarpectively, IS e OrV+e), @ lelX el 0, co ©, = AtVHT). @ 2 lease] Ie) are] PR ee] (es 8, = oo res IDI a atadl Pere i] lec aie] papel C23 6, ee wie Sule lewrelel ler” jaeae] “leiP re Pe po F200 o,® £0, os ©, = Ces 6, = les &, ei ey VPiayeKen B lose 8 v.d= apae), | Gris cf-ae]=4 2 (tte), iy a) 3 > eis vat)*+40 ’ | Paatet+ia= Vera)? #40 DP At = VA ErEG Hd => At6 = Vat44 444 |> A rardse = 37 : > 3654 TTA t4 §, wikis = $A=% 46017, 18,19, IZ 18 BS A=) ; 9928 ty 30,04, ao & 28, re : fio One og Sa £ GE re wlang; 43 Also, we Anow, RES " ‘ $e 1003 @.¥) uae 34,4) 10 1818 a) L-1@.¥) ‘ Se 10.2@.b= wo Eee lelIwlere=0 alll sino =0 «Seagal Soar iy cL oe e. OF Vowel ne el ss28 eo eae At Some Hme a ¢ BP can noe ve Ig Itt So, cites laj=o a Lo > : Eee $)| tonyerse vet alioays Tee eb 0, - 1a\ |B") sing = 0 Deiter Iwl=0 et Wl=0 of sino-o Sin 6=0° > 60° => rors phooluct x crbbivraX Vertes ale zo petaget Sahzapae => o ta Ae * ee ett ie i | 2 IWl- Jose trsm2o =] ip = heyy ; Fao figuss, x= we | yesin & | \ L24m? tn%2 J} | ==) _——— Dts Dingpnel Necth C= P+b”j a ey | Bey seer “m Oe a0 S| a tg 5 _ te Loaner 3 ven A * = 7 (22) “9 ew +e 09 “a es a 4 | > tant sgl (hal | le.vl= Vaatt edt = *fa4y | aoe, he / a) rivers shot | Dp = aa iv Og = wD - ae | Peo lid (externallay) | ees wo 6@ — 1 OF _ -307 0-sb | ™- =) | = 3e 450. | Midpoint of the tint Segment RQ, + nny — i = OR + OG t 2 = 30745040" 30" Mae : = 404900. oe posing ~ a SSS , AP OR the soicl— pont ef Mina segment RQ. " Romer Reoveo) : —_—— f 2 & 8)| tniven, Ae = 404+ x4 +tk B= ote aptae io? & 9 ot fs i ee 3 a —_ ke ~ 36: 2 a) .. oe fmmon Pein} a"he parallel +o Be pitied tha Sie colina _ ome BB dliviolds AC. im Raho fat a ¥ xX tes @= = Bes = 2 4 cr @) r= Me pige= 2 | @) yes ; 4X = x1 449 = 4 307 +Sin 305 = Ser —— 3d) OF datpy94z,0) 5 pot) | | OF (x,9+4,3 +240) oF Pe =0@ - OP a | swt ayy teres? © | WGI = J fen + bay)? +E,-22" - Sealer comporam ts = (23-21) s¥y-4,) Ey-2,) —— | Corsioles 3 vector Yio inmoia tg! 1s= Le) eiplsle7l |= IP) ste Fal 5) Givers thot, (AE PAE) & 0 uni} vectos = xt aapaxtle) 7 > Jr =) pe sina n@y