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veciors & $calers ) GE AB A et Beat Ee IF AN BF pe Sees A MA Ryd ed ah Pe DS, Te Jn Fe Pia ee Bet Be Pe ee ae ete DO Ne tgiee ‘ee “physical quamtity ushich Anas a emagmitude, but me spaifre diaection. €q.- distamce,, mass , Spead- The physical quarstity which Aas a crnagmitide, as uxil as disection, amd fellows dhe vector Lou & addition €g- Foxce , velocity , displacarment , mMememntum. Note -4] Cusumt Js met & vector quamntity. sheugh st Aas diaection asnd magmitude. as dt does mot “fellaur ~ vectou Lous K addition. A vectete is sasentated by deawwing am apsour prepetional inu Lomgth te the physical quomtity. being sapueSemtec.. im A vecten, vasdable Js suphsserted, by Om UVoLOUF Over english, Oo Faw alphabet. W The Aarme habet witheut dhe vectero sign wasents Sne emagmitude, & dhe yectes amd dhe Same alphabet with a Cap” Signy sapsesents the AIRACH ET & dhe Vecteto. Pacoctice. Tinne ee ons Ce tam (4B) « Te add dwe vecters , USimg dtvam four, place dhe toil J tne Becond veces on dhe Aead ok ‘che fist vector. Neus Complete the dsdam amd dhe sasultamt vectes Js given by dne aro Stasiting Jeon tad & the Fins vectou de the Aead ef Ine Aecond vecteu. = \e ges ¥ & 4. Bt vecto~ pee er ANGLE BETWEEN Two rer The a betwearns duse vests us defimed as the men- amg, ee vectes Uthem diney A anti dail de a i By - The asta. beluearry dhe vectors JS 60. s b iS ear 6 Led = NES a) ean ae eke ee Weo-Ace a pa rarer ADDITION OF \ECTORS ieee, When we Considers Swe vectess @ amd B soprusemted for Finding doe sasultant (+8) Using dhe Usdamge Lowe as Aheusn im fig. Usieng- pythagexas cheekiny Cl = J (arb cose + (bSimoye 11 = Jabr® cede + 2ablese resis lCl = Jor + + 2abcese amd. diewcten Js give by. o = dom! (_bsime +b ese, Paoctice Time =e: nt ) ) ) ) ) J gesultamt = 125 +254 50x1. = aq 50+.50 de = |50c@ +50 Na Veta) - @ = der! (es) | A Mam, Loalks IO Kr dewards east & dhew ametnes loRm dowards oo Neth east . Fimd Jts vesultamt g dinectHen. => C= 00 0Oxt A2 + 2 ne Fr = 506 Ae = 1043 ae 1O RM G= dom (543 ‘ 1o+15 | cee Sie | ener ee: | & = 30 | | | > IPARAKKEAOGRAM JAY OF or ag Hee ue place ithe vecter’s dad ite tail and Ccomphate the pasalldlogran Nour the gusubtant Js Grems by diagonal, & the poxallels quar naa ny dhe commen doi NOTE-4 Fe, omy ivery vectors @ , athe Symbels @ amd 11 mean ereactly, tthe Aarme hig. >____[PonyGon dA -OF ADDITION. |< Hes, we joier all the vectors te be Uaddeds im a Arend te tail Config Wrotiens amd mou the 2usuttamt Js givers by. dhe vector jeimi mf Laid f dhe fins vectes ie tthe proads of the ast vectow. c oY oS fe CA sapented, opptication of Tiare Lass) [NEGATIVE OF A VecTOR Negating Cha yectey JS a ? : een can aa vectete with dhe Zame mmagmritude but witty Eg - eb el [SUBTRACTION OF VECTORS Subtuaction, vs Same as the addition, ef de negatives, wy ere en i Oe OL i: Gl Ok Ol Gal ai a a oa a PRS Darn eee ire ly je ewe) lure on aoe eee CR ee ORY Wnt oe eRe ee er ee nn ee ee an » ~ (2) |} te dnowad = loox4 = Bon P | | Lite thuad = loo x 3. = Gon | | | | Z | tooN | | | | | | | | | | | | | | | | | | | An \ecteo, say 2 deausr, um the ve | ey Plame cary alusaus be useitters 43 | im Sasems of Sts Cemponemts || | te the % amd y-ands vies pectively. = ee vs > Fen @g - Sm ithe shown fig; e -f = ; @ Cam be ustitterrs as | = a Sucho oe ay) - | supresentation us catted Cadesian Aupaesenstalion . | | i} s +4 be | Owns Na + 3} (2) TB = -loces 3ch - 10 sincf =-5C- 555? | | | | | | a | 1 | | | >_[Poaar REPRESENTATION OF Vectors] | When g Vectory JS Apecified dm Lesems & dks magmitude amd | ome which dt makes with dhe x-ands , we Cal St palase | supresemtation,. | | 0 ee ee eee ee > > fConverrine FRoM fbLAR. FoRM To CARTESIAN Poem ees Considety a vector R raving, a magmitude R emakRing arn ame. 8 witty the x-ands as ANeuIn.. par \4 ‘ ' * Bo =RcCHed+ R Sim ef we ee RGesO B Represent che Shaw vectew inv Casctesiasry fourm $ 6 P= -5t +5455 40 = 10 Cesiaot + lo Sim 120f of PRINCIPAL RANGES FoR INVERSE Fun Sim! = -90 48 90 dam! = -90 de go Ces' = 0° de IBo >_ [Convertine FRom CARTESIAN FORM To PoLAR Form < Comsidew 0 ecto @ , as suprasented, dn casctesiam, {gum coe at tayt as Shown im 4g. @ ey G = dazeay Fens tame = oy | Lx O= tai! (23) Jf A, dS Positive a ae a ~-\ a . ° . © = 180 + tamd (94) Jf a, is megaline Comvesd the followin Cartesian te Felax fowms ; (a) 3t+4f = s@%e (b) -4t+ 3h = 5@ 143° () -3t-4f = 5@ 233° d a _ at = ae we -27° G) At - 3 5@ tum'(-5_) = 5@-37 EE DE Ee TO aE Oe a Oe Ee ee ee ee ee ee Oe ee ee ee ee ee ee a ee ee The Desitions vecis Ck @ Poirt im Space Js defined ar dhe vestor Asimimg dhe ckigim de thal Point. Foo eg: y= G+ Af+R >___[Maanrube OF A VECTOR _IN ieee Loo A 8B oc? = ob +8 ; 08? = opt+ AB F oc* = oft+ A+ BC C Of = o@e ccs tl * oe ° E We. fae ray + 0% G D DIRECTION CoSiNE OF A VECTOR The, Cesimes ef dhe am that a giver vector makes with xy A amd 3 axis saspectively, B Cesa%, CesB amd Ces (asw called ditections Cetimes. - c ACEO Ahab ea ZCEO =9¢6 fe) a E Cos d= OF oa UN ole G D Sake = Ox | a +05 +05 ay CaeB = p a Ces = Gz Q Cot t+ CoP B+ Caf y = 1 Sree + sinp + Sintys 2 De Mu pocan ON RECO By TSRRLE j =. Whemevese a vectors Js anultipvied by @ scalass , the mitude, ges mmultipGed amd dhe disection, © searmams game Jp dt YS @ positive Acalas, and, diswction sonenses Jf scalosy JB Megotive . s 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | I Zo )) ) ) ) Se ee es es es es Ba PF Se 2 ee oe ee Eee ae y) } ) J ) } POP See 3 or cee ree ) rr) a Pa k= 3sec, Pit = IQ Nsec: east = - IR Nsec. west Sf w posticle moves Fee one pesitior, Le ametheu then the Chang dry its petitiery vectey Js Called displaament vecterr. B= Ky - Fa Sy = fine pesition vectou 3 BH, = imitial pesition vectou y 1 By: g = lat-4f-SR x d d > => _ » a y SG = AY +3R > Beg = vat : > B= Rp - me = at -4p-3R Pret re det thew be twe vectses @ amd B Aaving an amg © belweary them Thers Wwe define dheit. pasduct as B A= abcese Fou eg: e “4 @.6 = 12 3 eel) 5 Dor _PRopucr iN CARTESIAN FORM Tat Sed Taf =.0 Tf 24 FR =O R.R 24 Rk. =0 2 ae i Det product us distdbutive ie. BV (B+e) = V.B+ v0.0 det drene be we Peints in Apacer , Aasnd B ushese. position vectexs axe Givers by Ry amd keg dhems dhe. Position, ef B ab Seem fem, A ds defined ab vectors dsining A te B uhich is alse Called pesition, B salddive te A (%gq). As itis clear fuer dhe figure position, of B uel is ae — Gy oS hn i R= Qtr eap+ear Bin = BF = Sq = B+ ap i A Bag = 2p -2R angle betwee GA & GC ee ie eral Ces@ a Q@= 60 ey Find. cafe b/ur body diagonals ok cube. ac cai at + aps ak GB = ef+at-en o.Ge = HHH ods A&B % ZA x ) ) ) ) ) ) } } _ dees Fe Fo F* SP I a eee ee ) J ) ) ) oD te Ae es ee es ee 2 | ae } y) ) ) ) ) ) ) }) ) ) re: ) J a, ) Ce ee ee Me Kee IG ee Ge eee ee age te Ge a Mtoe Me aN Ge) meee) om Pe Ral © Dee ee aa: Da] en coe ee eaten oe eemme Bes i’ NOTE + Sag, = GA (Staxdting fusmn G te A) | > _[Paarection OF A VECTOR ON ANOTHER VECTOR (CouPonenn}| jek dhow be twe vecters Zand B suprasented by Lengths OA amd OB, then the pupjections ek Aen B is Obtained by jeining the tus apecs Aen taig ds taiG amd dsepping- ray pespemdiutas, feem Ahead & A on vectes B (Say pt. A). Nour dA! we Lhe pusject- ion. As ot de Clea fom the frgwee , Pagiectiorn ef & enb oO a Ces a(S 1 oy eb o| ot " 1 a a o> ool det thew be tue vectors @amd Bb , hanine. am amgle 8 between them , then these cness product © is givers by —7_ Psp C=0x6 amd magnitude iS givers by C= ab Sime amd dimection is given, by sight Aamd dhumb sule oe the sight Anamad oe ae J 3 Jeiry “the vectex.s Je be Cuessed toil tte toil amd eotate a Scrat at the junction ros the Cemmmen tails Aug fiast vectes Je the Aecend vector chs the nen-szehlex amgle. the dicection | Ck the Aetabion, ef dhe Acuus is dhe dixaction, Ok CHugss product 4 | | | | | | | | | | | | | | | | | | | | | | | | | | | Similady , prajection of Bon = Ba | ) | | | | | | | | | | | | | | | | | | | | | | | | | | Place, the palm ef dhe aught amd pexpemdiculass te the palm ft dhe vecters Sud dhab the Littke finger ts Aleng dhe first vector amd tthe cud of the fingers peints dewaxrds the oe vectsty Thusugh, the nem- soflex amgle. New the thumb gives the Airuction, of Ine Cuess pueduct. ite dimection. |Lami's THEOREM | CONDITION FoR EQUILIBRIUM: A bedy Said, ds be Jin equilibsium df dhe Vector Susm of all the ANCES aching. on it HO. ) J2IJ5)J° 07) ° 31 2)2)7 7 SINE. AAU: Fd eer eee ee On 7 SinA SinBo Sing Ruse + . ~~" AP= ABSinB , AP= Acsinc © b ¢SnB = bsind G = _b Sine SmB 8 Piaie STATEMENT OF 4AMIS THEOREM ] | E | | | Q | | | | | | | | | | rw) 5 | REE epee ee relpcbe th equitibsium umdex, dhe influence % hear Jexces Fa, Fa amd adage Wail ws tail ther ; 7 | Sino SinB Sine ad | at Dp S Hesultamt fowe Js O, | ClegecL tbdamate Will be foumed, hee is SinA SinB Bing | F; | Sr = Fe me Re ire (B0O-A) Sin (ieo-B) Sm (185-Y) Fa = Fa = fe a: 1 Sina Sin B Sin a | fe 4 4 | a 4:5 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ice FRR Xe Fa | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | loon fa) 2 ne = ist) so 0 O86 e aig 9 lol © r 15 i nwo} u pe is ° ite) x 5 xb : Ele = a 
