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°.