Question

What are the components of the vector between the origin and the polar coordinate `(-2, (3pi)/2)`?

Answer 1

`(0,-2)`.

Explanation:

I suggest to use complex numbers to solve this problem.
So here we want the vector `2e^(i(3pi)/2) = 2e^(i(-pi)/2`.

By the Moivre formula, `e^(itheta) = cos(theta) + isin(theta)`. We apply it here.

`2e^(i(-pi)/2) = 2(cos(-pi/2) + isin(-pi/2)) = 2 (0 - i) = -2i`.

This whole calculus was unnecessary though, with an angle like `(3pi)/2` you easily guess that we will be on the `(Oy)` axis, you just see wether the angle is equivalent to `pi/2` or `-pi/2` in order to know the sign of the last component, component that will be the module.