Imaginary number vector rotation 60 degrees

Question

With which complex number can I rotate a vector 60 degrees clockwise?

I do not know where to begin I tried using the unit circle but I keep getting confused. Can someone help me?

Answer 1

Basically, in the polar form. A complex number is written as $z = \rho \cdot e^{j\phi}$ with $\rho$ the absolute value and $\phi$ the argument.

A multiplication would be $z_1 \cdot z_2 = \rho_1 \rho_2 e^{j(\phi_1 + \phi_2)}$ Knowing that, by using $z_2 = e^{j\phi_2}$ we can rotate any $z_1$ on a $\phi_2$ degrees. Note that $\rho_2 = 1$ to avoid changing the amplitude.

Finally, if you like the cartesian form notation, $z_2 = e^{j\phi_2} = cos(\phi_2) + j sen(\phi_2)$ would be your complex value.