Suppose I have a two-dimensional vector quantity in the time domain $\mathbf{a}(t) = [a_x(t) \;\;a_y(t)]$ where $x$ and $y$ denote orthogonal coordinate axes directions. This vector can be represented as an arrow from [0,0] to the point [$a_x,\;a_y$] at each time sample. The angle between the x-axis and the vector is given by $$\mathrm{tan}(\alpha(t)) = a_y(t)/a_x(t)$$ where $\alpha(t)$ is also a time-dependent quantity.
If I Fourier transform each vector component: $A_x(\omega) = \mathcal{F}(a_x(t))$, and $A_y(\omega) = \mathcal{F}(a_y(t))$, then I have a new vector $\mathbf{A}(\omega) = [A_x(\omega)\;\;A_y(\omega)]$, which is a function of frequency ($\omega$). This vector can also be represented as an arrow from [0,0] to the point $[A_x\;A_y]$ at each frequency. The angle is given by:
$$\mathrm{tan}(\beta(\omega)) = A_y(\omega)/A_x(\omega)$$
My question is whether there is any relationship between $\alpha$ and $\beta$?
For example, if the vectors have an angle of 90 degrees in the time domain will the vectors also have the same angle in the frequency domain (e.g. $\alpha = \beta = \pi/2$)?
If not, is there any mathematical relationship between the two angles?