H1.3) (25pts:5+5+[15:5+5+5)) (a) Plot the following complex numbers in the complex plane: ... ....x/ 1'. J93 1. (1) 1+ 2, (11) 1 z, (111) ? + is, (1V) 7 is. Now express (i)(iv) in polar form. (b) Express the following as complex numbers of the form .1: + g: 1 122" (i) 63%, (ii) (1 a3, (iii) (iv) cam/3m + a). (c) In what follows, we will use our knowledge of the manipulation of complex num- bers to prove an important result also arising in vector calculus. (i) Show that | 21 + 2212 = 12112 + 122/2 + 2Re(2122), (1) for two complex numbers z1 and z2. Note the z* notation denotes the complex conjugate and |z| = vz*z denotes the magnitude of a complex number z. (ii) By completing the square of 121 12 + 12 212 show that (1) can be rewritten as 1 21 + 2212 - (121/ + |z21)2 = 2(Re(2122) - 1211/z21). (2) (iii) Finally, using the fact that |z127) = [zillz2), deduce the inequality (3) In deriving (3), we have arrived at the famous triangle inequality, which states that the sum of the lengths of any two sides of a triangle must be equal or superior to the length of the remaining side. (Note that this result holds equally if we replace z1 = x1 + zy, and z2 = X2 + iy2 by vectors Z1 = X1 + y1 and z2 = X2 + y2)