Question

Find the modulus and amplitude for following complex number: (1 + 2i)² (1 - i)

Answer 1

Let z = (1 + 2i)² (1 – i)

= (1 + 4i + 4i²) (1 – i)

= [1 + 4i + 4(-1)] (1 – i) ….[∵ i = -1]

= (-3 + 4i) (1 – i)

= -3 + 3i + 4i – 4i²

= -3 + 7i – 4(-1)

= -3 + 7i + 4

∴ z = 1 + 7i

∴ a = 1, b = 7, i. e. a > 0, b > 0

∴ |z| = \(\sqrt{a^2+b^2}=\sqrt{1^2+7^2}=\sqrt{1+49}\)

= 5√2

Here, (1, 7) lies in 1st quadrant.

∴ amp(z)  = tan^-1\((\frac{b}a)\) = tan^-17