Find zw and z/w. Leave your answer in polar form.z = 1 - i w = 1 − 3 i

Question

Find zw and z/w. Leave your answer in polar form.z = 1 - i  w  =  1  −      3    i

Sharon Dawson · Accepted Answer

Find zw and z/w. Leave your answer in polar form.
z = 1 - i

| z | = \sqrt{1 + 1} = \sqrt{2}

θ = \arctan (- 1 / 1) = \arctan (- 1) = π - π / 4 = 3 π / 4

we are on the second quadrant.

z = \sqrt{2} e^{\frac{3 π i}{4}} = \sqrt{2} (\cos (3 π / 4) + i \sin (3 π / 4))

w = 1 - \sqrt{3} i

| w | = \sqrt{1 + 3} = \sqrt{4} = 2

θ = \arctan (- \sqrt{3 / 1}) = \arctan (- \sqrt{3}) = π - π / 3 = 2 π / 3

we are on the second quadrant.

w = 2 e^{\frac{2 π i}{3}} = 2 (\cos (3 π / 3) + i \sin (2 π / 3))

Hence

z . w = 2 \sqrt{2} * e^{\frac{3 π i}{4}} e^{\frac{2 π i}{3}} = 2 \sqrt{2} e^{\frac{17 π i}{12}}

= 2 \sqrt{2} (\cos (5 π / 12) + i \sin (5 π / 12))

z / w = \sqrt{2} / 2 e^{\frac{3 π i}{4}} / e^{\frac{2 π i}{3}} = s q r t 2 / 2 e^{\frac{3 π}{4} - \frac{2 π}{3}} = 2 \sqrt{2} / 2 e^{\frac{π i}{12}}

= \sqrt{2} / 2 (\cos (π / 12) + i \sin (π / 12))

Find zw and z/w. Leave your answer in polar form. z = 1 - i w=1-sqrt(3)i