Question

Write z1 and z2 in polar form and then find the product z1 z2 if:
z1=2(sqrt3+i)
z2=3(1+i sqrt3)

Answer 1

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 < angle < pi/2). sine(angle) = complex_part / magnitude
sin(angle1) = 1/2 => angle1 = pi/6 (30 degrees)
sin(angle2) = sqrt(3)/2 => 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.