Complex Numbers 20 2122 01+62 0 Flgure 1.11 Geometric interpretation of the product. The abbreviated version of Eq.(7)reads as follows: 21z2=(ri cis6)(r2cis)=(rir2)cis(+) and we see that The modulus of the product is the product of the moduli. kzizl=lal ll (=rr2); 田 The argument of the product is the sum of the arguments. arg2132=argc1+argz2(=91+62). ② Tobepcise th ambiguous E()is tobe interpreted as saying that if particular values are assigned to any pair of terms therein,then one can find a value for the third term that satisfies the identity.) Geometrically,the vecor has length equal to the product of the lengths of the vectorsandand has angle equal tothe sum of the angles of the vectors and 2 (sce Fig.1.11).For instance,since the vector i has length 1 and angle/2,it follows that the vecorca be obtained by rotating the vector through a righi angle in the counterclockwise direcion. bseving thai diviisthertioopctionertohe following equations: 号-g一+in份-=份 (10) g()=g1-g, (1I) and kil l22 (12) c80 of the lengihs of the angles of the vectors and z2 and has angle equal to the difference of the vectors and z2
1.3 Vectors and Polar Forms Figure 1.12 Perpendicular vectors. Example 3 Write the quotient (1+)/(3-i)in polar form. Solution.The polar forms for(1+i)and(3-i)are 1+i=1+il cis(arg(1+i))=2cis(/4). √3-i=2cis(-π/6) Hence,from Eg.(10),we have -孕-（别9留· Example 4 Prove that the line/through the points and is perpendicular to the lineL through the points z3 and z4 if and only if g名二器=号 (13) Solution.Note that the lines!and L are perpendicular if and only if the vectors -z and 23-z4 are perpendicular (see Fig.1.12).Since arg 21-22 =arg(1-32)-arg(z3-24) °3-4 gives the angle from3-4to-2.orthogonality holds precisely when this angle (up to an integer m ultiple of is equal oor.But this is the same as saying ihat(13)holds. Recall that,geomneirically,the vectoris the reflection in the real axis ofthe vector (scc Fig.1.13).Hencc we see that the argument of the conjugate ofa complex number is the negative of the argument of the number;that is, arg元=-afgz (14) In fact,as a special case of Eq.(11)we also have arg-arg Thus and have the same argument and represent parallel vectors(see Fig.1.13)
Complex Numbers 2 Figure 1.13 The arguntent of the eonjugate and the reciprocal. EXERCISES 1.3 1.Let z =2-i and z2=1+i.Use the parallelogram law to construct each of the following vectors. (a)z1+2 b)21-z2 (c)2z1-3z2 2.Show that azza=zz3l. 3.Translate the following geometric theorem into the language of complex numbers: The sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of its sides.(See Fig.1.6.) 4.Show that for any integer.(provided when k is negative). 5.Find the following. a刨/￥2x 2+i)3 -2-i (b)(1+i(2-3i)(4i-3)(c) (1-)2 @画a+)0w1 (-i)100 6.Draw each of the following vectors. (a7cis(3r/4)b)4cis(-π/6)(c)cis(3π/4) (d)3cis(27m/4) 7.Find the argument of each of the following complex numbers and write each in polar form. (a-1/2 b)-3+3i (c)-πi (d)-23-2 (e)(1-i0(-√3+i) (国(W3-i)2 ®+ 万+i 8.Show geometrically that the nonzero complex numbers z and z2 satisfy + +if and only if they have the same argument. 9.Given the vector interpret geomeirically the vector (cosi sin)z
1.3 Vectors and Polar Forms 23 10.Show the following: (a)arg z1z2z3 argz1 arg z+argz3 (b)arg ziz=arg z1-arg z2. 11.Using the complex product (1+i)(5-i)4,derive π/4=4tan-'(1/5)-an-l(1/239), 12.Find the following. (a)Arg(-6-6i)(b)Arg(-)(c)Arg(10i)(d)Arg(3-i) 13.Decide which of the following statements are always true (a)Agz1z2=Agz1+Arg2ifz1≠0，z2≠0. (b)Arg=-Arg z if z is not a real number. (@)Arg(1/z2)=Arg21-Ag22fz1≠0,2≠0. (d)argz=Agz+2rk,k=0,±l,土2.，ifz≠0. 14.Show that a correct formula for arg(x+iy)can be computed using the form [an-10y/x)+(π/2)[1-sg(r1i迁x≠0， arg(x iy)= 红/2sgn0) ifx=0andy≠0， undefined ifx=y=0. where the"signum"function is specified by [+1if1>0, sg()= 0if1=0. -1ift<0. Show also that the expression sgn(y)cos(x/+y2),at its points of continuity. equals Arg(x +iy). 15.Prove that lz-z+lz2l. 16.Prove that llztl-z2l1-zl. 17.Show that the vector zt is parallel to the vector z if and only if Im()=0. 18.Show that every point z on the line through the distinct points z and is of the form z 19.Prove that arg=arg z if and only if =cz2,where cis a positive real number. 20.Let z.,and za be distinct points and let be a particular value of argl(z3 Z1)/(z2-z1)1.Prove that k3 -222=1z3 -z2+l2-z-21z3-zllz-zcos. [HINT:Consider the triangle with vertices.3.]
24 Complex Numibers 21.If r cis =rcis+rcis,determine r and in terms of r.r,01.and 02. Check your answer by applying the law of cosines. 22.Usc mathematical induction to prove the generalized triangle inequaliry 会2ar 23.Let m,m2,and m3 be three positive real nunibers and let 1,32,and z3 be three complex numbers,cach of modulus less than or equal to 1.Use the generalized riangle inequality (Prob.2)toprove that |1Z1+m222+m323 ≤】 m1十m2+m3 and give a physical interpretation of the inequality. 24.Write computer programs for converting belween rectangular and polar coordinates (using the principal value of the argument). 25.Recall thal the dot (scalar)product of two planar vectors v=(x1.y1)and v2= (x2.y)is given by V12=1+12 Show that the dot product of the vectors represented by the complex nunbersand z2 is given by 21·2=Re(1z2). 26.Use the formula for the dot product in Prob.25 to show that the vectors represented by the (monzero)complex numbers and z2 are orthogonal if and only if ztz2= 0.[HINT:Recall from the discussion following Eq.(9)that orthogonality holds precisely when =icz for some real c.] 27.Recall that in three dimensions the cross (vector)product of two vectors v:= (x0)and v2=(.0)in the xy-plane is given by y1×2=(0,0,x2-21). (a)Show that the third component of the cross product of vectors in the-plane represened by the complex numbers and z is given by Im () (b)Show that the vectors represented by the (nonzero)complex numbers t and allel if and only if Im( 0.[HINT:Observe that these vectors are parallel precisely whenfor som alc. 28.This problem demon ysis of planar mechanisms. rates how complex nolation can simplify the kinematic anal- Consider ed-pistoninkae depiced ni1.14.Theaa a engine,conibustior forces would drive the piston and the connecting arm bwould transform this energy into a rotation of the crankshaf.)