Question

Given w= sqrt2 (cos ( pi / 4 ) + i sin ( pi / 4 ) ) and z = 2 (cos ( pi / 2 ) + I sin ( pi /2 ) ) , what is w – z expressed in polar form?

Answer 1

w = √2 (cos(π/4) + i sin(π/4)) = √2 (1/√2 + i/√2) = 1 + i

z = 2 (cos(π/2) + i sin(π/2)) = 2i

Then

w - z = (1 + i) - 2i = 1 - i

so that

|w - z| = √(1² + (-1)²) = √2

and

arg(w - z) = -π/4

In polar form, we have

w - z = √2 (cos(-π/4) + i sin(-π/4)) = √2 (cos(π/4) - i sin(π/4))