Unit-2 Vector Algebra PDF

Kechavi Mekta Page Dale: Mcn T ue Wed Thu Uit-2 Vector Algeba Canteut Al...

Kechavi Mekta Page Dale: Mcn T ue Wed Thu Uit-2 Vector Algeba Canteut Alqebruc Operations Basi PaLopauties aud ineg ualities Distance aud oms Applications : Velecits and Aceele> atiou Dekinition Vector Aquantite ttrat hat has has magurtude as wolla direion quantity Rs calleda 'vec tor Notatiou: a, AB Desinition Bsitfo Vector Fo a point P(xi Y,)in 3-D Coobdiuate system, op os is the position veotor -bitth as inita pont tlagitade ( Directiou Cocines ad Drectioy Ratios. Position vector DP makes augle oei B aud x woith Dugle , Uy and '2 ares Co cosoa, CosB, CoSY CoSY ate dine ction 'cosine o Vector OP aud denotd m, n  2 Atco 0 m ? (Oso = cosp ypes 4 lectors (DZero Vector: ternoinal poins whase aud Bital Vertor A vector Called a &euo s calledl coineide,s Vector with maqaitude zero ze4O Vector dencted t Veotor imagnîtide unity Ge.J un?t) A vector whose vetor:g the itVectorTue eit Re called a Veetor a' Pc denoted diyectton A Cointfal vectors came 9u4fal mobe ectors haoiug the Thoo o coiitial dectors Poînt ae callet Colliuea Vectors: T0o O) mobe Veotors ale datd to be coltiuoas ff they ate paralel do the ame line , iespoctive of their maguitude aud diections tqual Vetos:  Fage Date Tuoo e mOse Vectors a a nd b aste sid to be equal f they hawe the Same maqitude aud direction vegarlless of the pocitiobs ot their Tnifal potts aud Negative of a Vector A vedorwhoce maitude Ps the sane as that of c give voctor but e dioction e cPr&site to it, ise cated nejatove of tbe vector Atditicu of Vectors t. Triaugle Law o4 Vectar Additisu = AB+Bc Aß + BC- AC = o A AB +BO +cA lasalleleqram Lao ct Additieu OA OB A Components of a vector Teco,0,1) et us take the point A(,0,o), B(o,1,0)..cCo,0,1) Bto,t,o on the X-ais, H-aris ACI,0,o). laud 2-ais vespecfely Then clealy Veotors: OA OB aud D each haue aye allo d uit ve ctors the qxo's OX aloag  Tecpecthyely and donoted as ,fpu(0,0,J Hence (1,0,o). 1.: o:t.0). of positon ector opvec Noco, cons ides the pociton vector tor poiut PoaH3) Then Poit eheronge qrien by ak cald com pouent of anu Vector fs ccala Ibis horm called the bom tete and 2 cue of aud i Êand 3k a compouets called vector componorts A The lengts by. Some important results A A britbe tbg = a/i +0>t4g k and b = a 1hen A a tb =(aitb)? ) 3) he wectos d'ayd bate equal i7 aud ony if ) The mal tiplicatiow 6 veclor a by any_ scalar A fs given by  Jlcra foce: Date bt a and b be any too Vectors and kk and let m be ccalass tho katma = (ktm) a i) kOma) (km)a Fird the values of x yond 23 cothat the vectors thatthe e Qud equal Sor Too ectors ase Cqual aud cay if their corvespoudiug Componete qual. So 2, -23= ot ai+2Ë aud b= 2?+. T ä - [ ? Are he ve ctors a aud b eq ual 2 V5 s o a | : 1B But Hhese tao ve Ctors. ate not equal as tacÕr correspowding Componente Ole not Same ER FRod he unit vector n the die ction 6f vector 2i +31 The tenit vector în thediection of avector q' is ta'| Tal 4+4+) Now,  Thaekore VIg of vecfor a'=i-24 drecton E Find a vector i the ' wit that has maqntude dietion of given vector a So The wnit tector in the hacing magnitado eq ual to vector Tkenefore thethediracton aud ot 7a = V5 VG Ex Find the enit veotor in theolirections O the cu) 3 A L (2n (513) 4 4 3 - 9k Thenehore wit Woctor P V99  Fege |Dile : Produet of Teoo Vectos cala) cdot)product of too vectors : he ccalas product of to non Bero vectors a and b denotd by abic de hincd au a.b=loo shee Rs the augle betweon brtwccn OKoKJT az0 or b E0 hen 9 ic not de )X Resulb ( aeal nunbes () st a aud b too non 2e40 vectors: thau a-b 0 if aud only if a aud B ate pupeudiculd as to each oteu thona.b=lallo Tn pouttculasaa t&|! 49J then Gfo mtually peupeudieulal pesieud wit vector we haue  veotorc a aui G The auglebetween too ron zero qiien by Cos Procuet ic comiiutative The ccala, a ( Dictibtivity of cralas Product ouese additiou tcoo salass aud fs.aand be auy vector touay aud tb bst bak Then ab (ai A +ba) xI Find the Qugles betwcen too VectorS and o git magnitde 1 and 2 Given a. vespecfvey aud.  Page: Dale : EX2. Fixd augle &bekaeen the vectors-k E3 a 5i --gk and b:+3¡ -sk ten shea that the vecfors d o aud ab'obe poupendculea 6i +2i - + 2k -y.t2k) 24 -8 - (6 Hacu find lzl Hhen Since a is a it veetor:| = t' Also (B-a) (ta 8  tojection ofa vects adon cthe2 Veco. vegtor a =2143jt Ex: tind the projecton Proje+tk ofthe on the vectos on the vec tos b ic ton of a vectos a The phojec gien by t6t 2 (2;43j+2k) +2 tkJ = 2 10 t0o Vectors d and b aye such find la- bliH Qud a.L 4 at läl=2, :3_ both lule have li-bI2: (a-).a-6 he haue (a|2- 2(at)+1012 y- 2(u)+9 for any too vectors a and b, la bls lalE  Face : |Daie: @ Trionale Tnequality: Foran oo vectors d ond t àtbKlol4bT EX VÙsrify Cauchy Schoaut incqualalH cud A k and b31- i + 2k Vector or Cross Procuet &Two vectors: The vector Product :of too non zero Vectors a aud is denotecd a X ax b = wheste is the amgle ketoeen a a and b 0x0s and nn ie a wit ector peufeu diculas to Results :(O axo is a veoto (2) at a cnd b be too non ze40 Vectors, Then i and only if a audh 'ane poallel (os' colliuea) to each othei then a xb= |aIRI 2  Thu 0 Find wut taugent vecfor to he The pocition vecor bf a pastcle ic 'z(t)i + Asin 3t; + &cost 3tit 8tk Velocits =V= dst 4d(2cin3t)} +d (Dast1dE dt dt kcos2t- 6sint 34j+ 8t : lvl= 86cos3t +36 sin'at +64 i 36+64 - 1o0 -1o d =-18 sin3t- Rcost 3+ dt Noo lal G'cint+te)cos'st 2