Vectors - Lecture Note

classmateDatePageVectorsThey are quantities with both magnitudeand directionLocalised vectors are those vectors for whicha point of action is defined For example, AB.Freevectorarethosevectorsforwhichapaintofaction is not defined For example, iCouinear vectors are those that are parallelor anti - paravel. Note that it a & 6arecollineara=sb&axb=0.Addition of VectorsThetriangleinequalitystatesthat-Tal - 151/ s a + b / tal + 151a+65ainequalitystatesthat-Thepolygon-Ias+a2+++anup lail + 1a2/+ las 1++ lan!a3Position Vector of a PointPQpos vec of P = P-pos vec of Q = qThus, PQ = q - p. vector OF,classicalis; have BODYDatePageofLPDA = 90(x,y,8)LPAD = tant(1)2)LPAO = 90o(4.00)LPCO = 90e(0.0,3)(x,0,3)iSection FomulaAcaP = mb+nainP(F'InBLE>AP 11 PgAP = >PB = ("M/PBi a = (m/w)(6 -p)up - na = mb - -mp10 = nb + namthCenouid of a TriangleA(x)4of - a + B + 1(97)c(r)3A (s)g - 21 FTF) txis = B+F32=18teach+F+Fgo = 2(5) + =33 PageIncentreofaTriangleA(2)CcI(F)I'B(R)D(5)c(8)aEcbc)aangle bisecturb+ctheoremLinearlyDependent&IndependentVectors-Thevectorsa+a,a3an are -linearlyindependentif>a+>2a2+r n an = 0onlywhenr=r2>n=01dependentifa+r2rhan=a2+0linearlyis possibleforatLeastvalueofrtO.Notethat-twocollinearvectors areLDcollinearvectorsareLIotwononri >2(2L)+ 2(1+j)vectorsareLDthreecoplanar2 & Dr, non zerovectors areusLI.threenoncoplanarLDr(o)=invectorisonezeroonezerovectorisLI .nonvectorsarealwaysLDoffourormoresetLIThusifa6noncollineathen,&are,x, a + y 6Dimpliesx4=D=y,=a=x2a+24+yy,bimplies24X2&4,=y2.6 The condition for colinearity of twovectorsis:aa ^ = 6a->a=r5 =or a = - -5parallelantiparallel.a x b = 10they are linearly dependentIfa=ai+a2j+askisparalleltobb,by k01/101=02/102=i03/122=+b2j+or a x 6 = 0 . -Twovectorsarealwayscoplanar,twopointsarealwayscollinear&coplanarthreepointsarealwayscoplanarTheconditionofcoplanarityofthreevectorsis:a (5 x c) = 0(scalar triple product)as a2 aza = air + anj + azkbb2b2=06= b,0 + baJ + 63kC C2 C3c = 40 + c2j + C2 kaxb=1 (b x c)they arelinearly dependent=noteIfP=xa + y.b + 3.cwhere a 5q=x,a+42b+32Cc are www1=x a+42 b+-33Ccoplanarp q&1canbecoplanarif5 tqr)=D, 14qF]=Dx y, 3box product - scalarX2 y2 32[a. 6.c=0triple priorX3 y3 32 DatePageTheconditionofcollinearityofthreeis :points(c - b) =(5 - a)= u (c - a)(c-b)x(5--a)=10(anypairs)ThesidesformedarelinearlydependentCg((E)AB&BCA(5)(a)should be-,/courinearorigina&care"sectioned"by6section fornuda.mainxa+yto+Zc=0wherey+=0expressionxa+b(-x-2)-+ze=0not all zerob =xa+zcsectionx+formula}Theareaofatriangleformedusinga-b&ciszeroTheconditionofcoplanarityoffourpointsis:ThethreeconnectingvectorsarecoplanarAB[ABACAD]= 0AC+ytoZC+wd=0wherexty++20=0xa+g+not all 0 PageScalar / Dot Producta. b = TallblcosoIt produces a scalar value and isdistributive, as well as commutativeNote that (a)2 = a a = tal2coso = 1212Vectors along Angle Bisectors5qP = > (a + 6)P(internal bisector)aq= s(a - 6)(external bisector)The corresponding unit vector along - P istax,where D is the angle between2cos%the vectors a & bThe corresponding unit vector along q is+ / a - 62sin /2ProjectionsaProjection of a along 6 isQ6lal coso = a.b = a.61b1and b along a isIbleoso = a.b = T.alal CLASSMALEDatePageVector componentsaThe vector component of -AQAa along To is (a.5) 60$ 6a perp to 6 is a-(a.5)6OA = 18 + PAPA=DA-OP=00=a1notethat(a.i)i.+(a.j)j+(aE)k = aVector/Cross Productaxb = 116/sino inItproducesavectorperptoa&b,/aswellastheplanecontaininga&bNote that :a x 6 = - b x aa x (5 + c) = axb + axck (axb) = ( ka) x b = ax(kb)a x b can be determined as follows-ijkax5 =ax ay asbe by byTheunitvectorperpendiculartoa&6isIblaxblnote (a. b)2 + (ax5)2 = + a262sin'to = ath PageScalar Triple Productsa (bxc) - [ C ]The scalar value produced is thevolume of a parallelopiped of sidesa, + 6, E=volume =cbbase area heightaNote that the dot & cross can beinterchanged. so a.(bxc) = b. (cxa) =c..(ax55 [] = [5aa] = [cab]Also [a b c] = -[a=5] (Exto = - bxa)= [aba]And[a+Toc a] = [aca] + [6ca]andkl a 5 c] = [ka 5.E] = [a kt= etcIf any two vectors in the box are parallelthe product is zero[5 6 a] = [a 5 15] = [5 a 5] = 0Then[a 6 c] = To 2]Note that[axb bxc cxa] = [a t E]"The value of [a 5 o] can bedetermined as follows - ax ay aza. . (5 x c) =bx by byCx Cy CzAlso-l.a l.b l.c[e in u]C. a b c] =m.a m.b m.cn.a w.b n.ccrow by row det. multiplication) Volume of a TetrahedronAvolume = 1/3* base area height= 1/6 [a To c]-ac0C5BVector Triple Productax(5xc) = (a.c)b - (a.b)c = xb + ycperpendicular to asame plane as I & c .(a b)xc = - cx (ax5)= [(C.T) a - (c.)bl.(-1)= (a.c)b - (b.z)aThus, a unit vector perpendicularto a butin the plane of To & c is I ax (6xc)la x (5x2)If a, b, c are non coplana vectors,ax (bxc), bx(cxa), cx(axb) are coplanarand form a trangle, since their sumis zero.If a , / b c are non coplanar a x b - bxcand c x a are non coplanar. DUTEYPageScalar Product of Faur Vectorsa.c a.d(ax5). (cxd) =5.2 b.d(axb). (cxd) = a. ( x x (cxd)) and x prod.we can interchange= a ((6.2)c - (6.2)d)= (a.E)(5.d) - - (a a)(5.2)Vector Product of Four Vectors(ax5)x (cxd) = [a Taja - [atc]]it x (cxd) = (u . d) a - (u.c)d(ax5) v = Ca.J)E - (5.0)a -=[a a d ] 6 - [bcd]aNote that the resultant vector lies alongthe intersection of planes formed bya, b & c,dAlso, (axb) x (cxd) = 0 implies that eitherall , alld or the planes are parallelSimilarly (2xb). (cxd) = 0 implies thatc 11 d or the planes are perp Cauchy Schwartz Inequality(ai a,2+ (bi + b2) > lab, +a2b2)2asfollows -It canbeprovedcosos & so a.b.sab, + a2b2 Vai+az b,2 + b2(a,b, + azbz)2 VI (ai + a,2 (bi + b2)where a = a,i + asj & To = b,i + b,jIn general,(ai + az+ + ++bn)2> (a,b,+ A2b2 ++ anbu)