Let A = a(cos a+j sin a) be divided by J5 = 6(cos ,5+y sin /5),Engineering Mathematics Foundations
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Section titled “Why This Family Matters”Routes the mathematical language Steinmetz used to train engineers in calculation itself.
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Candidate Formula Cards
Section titled “Candidate Formula Cards”If, A=ai +ja2 = a (cos a+j sin a), thenand ai + ja2 = a (cos 6 + j sin d) ;or ai -\-ja2 = A(cos 0+j sin 6).and A ==5 (cos 37 deg. H-j sin 37 deg).C = AB = ah (cos a+j sin a) (cos /?+ / sin /5)A = ai +ja2 = a(cos a+j sin a),20. If ai+y6i=6+2.5 J is represented by the point Pi;and the vertical distance 6i=2.5. If a2+jb2 = S+4:j is repre-Pi: a:=+3, y= +2 P2: x= +S y= -2,• Pi=3+2/ P2 = 3-2/ P3=-3+2/ P4=-3-2yPi = - -^a, P2 = - 7— a,V +l=cos qX + 7 sm qX - ,cos x+y sin X = ~1,a+y6 = 3 4-2?s, the point P2, 90 deg. away from Pi, wouldP2 = jPi-Ka+jb)=j(3-V2j)= -2+3/,izontal distances of Pi and P2: ao = ai+a2 = 6+3 = 9, and aao +jho = {ai +a2) +j(bi +62)a2 +jb2 = (ao -di) +j(bo -61)^ = a(cos 0 +j sin d),but since ai=a cos 6 is negative, -4, cos 6 must be negative,4 = 5 (cos 217 deg. +j sin 217 deg.)= 5( - cos 37 deg. -j sin 37 deg.)^0 = So (cos do +j sin ^0)isi,n 6i=j^ = 0A50, as ^i = 24.3deg.♦^2 = S2(cos d2 +j sin dz)/S4 = 1105 (cos 30 deg. +y sin 30 deg.)S6 = sq{cos Oq +/ sin Oq)AB = {aihi -a2?>2) +j{aih2 +^2*^1),For instance, A=2+j multiplied by B=^l+1.5 j gives= a6 {cos (a +/?) +j sin (a +/5)} :Thus, to multiply the vector quantity, A = ai+ja2 = a (cos« + y sin^ by ^ = 61 +J62 = & (cos /? +f sin ^) the vector OA in Fig.A = 3 +4/ = 5(cos 53 deg. +/ sin 53 deg.);(7 = 44 = 54(cos 4 X53 deg. +j sin 4 X53 deg.) ■= 625(cos 212 deg. +j sin 212 deg.)(7 = A/A = A”=a”(cos-+7sin-)A = a\(ios{a-\-2q7:)-\-j ^m.{a+2qn)\,C= ■<lA= i625(cos ^ - —+jsm 1 ^)= 5(cos 53 + j sin 53) = 3 + ij= 5(cos 143 +j sin 143) =5( -cos 37 +f sin 37) = -4+3/= 5(cos 233 + J sin 233) = 5( -cos 53 + / sin 53) = -3 + 4/= 5(cos 323 +y sin 323) =5(cos 37 -j sin 37) =4-3/in opposite direction from A, Inversely, if we take AC= -2,vertical distance, jBP= +2f, and therefore is given by aIf jx = \og, (-1), then £^’^= -1,Pi and P2: 6o = &i+&2 = 2.5+4 = 6.5, hence, is given by theper sec, making an angle ^o = 20 deg. with the x-axis; hence,24. If A = ai+ja2 and B = hi-\-jb2, are two general, orIf now we have two impedances, OZi and OZ2, Zi =ri -jxiIf we have a current and a voltage, I = ii + ji2 and E = ei -\-je2,For instance, let 4= -529-33iy = 625 (cos 212 deg.+but since ai = a cos 6 is negative, ~4, cos 6 must be negative,a2 = a sin atance AB, and multiply by (-1)^ we get the distance AC= -2and multiply by (-1), we get AB=+2; that is, multiplica-more by V-l, we’ get +2 X v^^X V~l= -2; that is,combination of the distances, 05= +3 and BP=- +2/. ForPs’ x=-S, y=+2 P4: x=-S y=-2]x= - = and 2/ = - =/; or, is represented by the general nuniber,log£(-l)=/7r(2n + l),jhk= +1.khj= -1,The only feature which must be kept in mind is that f = - 1, andf =/, f= -1, f = -], f- +1;f = +y, f= -1, f= -/,, f = +i;x+jy = 5-Sj,x = 5 and y= -3.the general number a+jh = 6+2.5j may be considered asdistance from the y axis, 0A = BP = a^6, and the verticaltan <? = :== =-77- fl/)^‘tan 6=-.