Write z1 and z2 in polar form, and then find the product z1z2 and the quotients

Question

Write z1 and z2 in polar form, and then find the product z1z2 and the quotients z1/z2 and 1/z1. (Express your answers in polar form.)
z1 = 1+sqrt3i, z2 = -2sqrt3 + 2i

Explanation / Answer

First magnitude using pythagoras' theorem. In both cases it is sqrt( sqrt(3)^2 + 1^2) = sqrt(4) = 2 Now work out the angles. Both terms are positive so they're in the first quadrant (0 angle2 = pi/3 (60 degrees) To multiply, you multiply the magnitudes and add the angles. To divide, you divide the magnitides and subtract the angles. To do 1/z1, do the same thing, remembering "1" is magnitude 1, angle 0 degrees. So for example z1/z2 magnitude = 2/2 = 1 angle = pi/6 - pi/3 = -pi/6 => z1/z2 = magnitude 1, angle -pi/6 which is (though you don't need to do this) sqrt(3)/2 - i/2 Check with normal division (again you don't need to do this): z1/z2 = (sqrt(3) + i) / (1 + i sqrt(3) = (sqrt(3) + i)(1 - i sqrt(3) / [(1 + i sqrt(3)(1 - i sqrt(3)] = (2sqrt(3) - 2i) / 4 = sqrt(3)/2 - i/2 Do z1 x z2 and 1/z1 in the same way.

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