Answer to Question #116823 in Algebra for desmond

Question

Find the moduli and principal arguments of w, z, wz and w/z, given that
(a) w = 10i, z = 1 + i√3, (b) w = −2√3 + 2i, z = 1 − i, (c) w = 1
1 + i√3,
1

Answer 1

a) |w| = 10, arg(w) = $\frac{\pi}{2}$; |z|= 2, arg(z) = $\frac{\pi}{3}$; wz=$10*2*e^{\frac{\pi i}{2}}*e^{\frac{\pi i}{3}} = 20*e^{\frac{5\pi i}{6}} => |wz|=20, arg(wz)=\frac{5\pi}{6}; w/z = 10/2*e^{\frac{\pi i}{6}} => |w/z|=5, arg(w/z)=\frac{\pi}{6}$

b)|w|=4, arg(w)=$\frac{2\pi}{3}$; |z| = $\sqrt2$, arg(z) = $-\frac{\pi}{4}$; wz = $4*\sqrt2*e^{\frac{5\pi i}{12}} => |wz|=4*\sqrt2, arg(wz) = \frac{5\pi}{12}$;

$w/z=4/\sqrt2*e^{\frac{11\pi i}{12}} => |w/z| = 2*\sqrt2, arg(w/z) = \frac{11\pi}{12}$

c)|w| = 2, arg(2) = $\frac{\pi}{3}$; |z| = 1, arg(z) = 0; wz = w*1=w=>|wz|=2, arg(wz)=$\frac{\pi}{3}$; w/z=w/1=w=>|w/z|=2, arg(w/z)=$\frac{\pi}{3}$