Question

Suppose z is a complex number (a point in the plane) with angle 30 degrees and radius 2. Suppose w is a complex number with angle 60 degrees and radius 5. If z and w are multiplied together the angle of their product is how many degrees?
Answer value

Answer 1

Write each number in complex polar form:

$z=2e^{i\cdot30^\circ}$

$w=5e^{i\cdot60^\circ}$

When you multiply two exponential expression with the same base (here, that's e), you end up adding the exponents, so that

$zw=(2\cdot5)e^{i\cdot30^\circ+i\cdot60^\circ}=10e^{i\cdot90^\circ}=10i$

so the angle of the product zw is 90°.