Question

Find the modulus and amplitude of the following complex numbers.

−3(1 − i)

Answer 1

Let z = −3(1 − i) = −3 + 3i

∴ a = −3, b = 3, i.e. a < 0, b > 0

∴ |z| = `sqrt("a"^2 + "b"^2)`

= `sqrt((-3)^2 + 3^2)`

= `sqrt(9 + 9)`

= `sqrt(18)`

= `3sqrt(2)`

Here, (−3, 3) lies in 2^nd quadrant.

∴ amp (z) = `tan^-1("b"/"a") + pi`

= `tan^-1(3/(-3)) + pi`

= tan⁻¹(−1) + π

= − tan⁻¹(1) + π

= `-pi/4 + pi`

= `(-pi)/4 + pi`

= `(3pi)/4`