Angle of attack from velocity vector
It is exactly the other way round:

$$ \alpha = \mathrm{arctan} \left( \frac{v_z}{v_x} \right) $$

You can simply derive it from the trigonometry in a triangle:

In this image the aircraft is moving to the right (along the arrow labelled v) and the $x$-axis is pointing along the aircraft body.

The resulting function looks like this:



For $ v_z = 0 $, you get $ \alpha = \arctan(0) = 0 $.
For $ v_x \to 0 $, you get $ \alpha \to \frac{\pi}{2} = 90^\circ $.