When rotating an object the vanishing points would indeed move accordingly, but the space between them would shrink or grow, depending on how you rotate it. The space between a set of vanishing points is smallest, when the object you are rotating is facing you at a 45° angle.
This becomes clear when you are trying to imagine an object in one point perspective, so an object at a 90° angle. One vanishing point is directly in front of you and the other infinitely far of to the side.
Keeping this in mind you'll want to move the vanishing points further apart, the more it is rotated from a 45° angle. If you don't want to guess, here's how to get it accurately:
You'll have to chose a station point which I marked as SP. Usually, the horizontal position of the SP aligns with the centre of the image plan, but I chose to align it with the frontal edge of your black cube in this example. Then, you'll move it down to the point, where the lines between the vanishing points and the station point would describe a 90° angle (green in the picture). From there on rotating the vanishing points is as simple as drawing a new set of two lines connected to the station point that are still at an angle of 90° (here in turquoise). The new vanishing points are where these lines cross the hoirzon line.
I you now wanted to know the accurate lengths of the edges of the object you'd need to have a perfect circle in perspective as a point of reference (which is basically any ellipse with the minor axis perpendicular to the horizon line that is sitting on the ground plane. Just make sure there aren't multiple different ones). I recommend just eyeballing the length of the sides.
