Final answer:
Neither vessel is considered the stand-on vessel in fog when they are not in sight of one another; both must maneuver cautiously and use sound signals to avoid collisions.
Step-by-step explanation:
The question pertains to the rules of navigation, particularly in conditions of limited visibility such as fog where vessels do not have visual contact with one another. According to the International Regulations for Preventing Collisions at Sea (COLREGs), specifically Rule 19 - Conduct of vessels in restricted visibility, the answer is that neither vessel is the stand-on vessel when two vessels are crossing in fog and are not in sight of one another. Therefore, both vessels are required to maneuver cautiously and use sound signals to communicate their actions. The rules emphasize avoiding collisions by adjusting the vessel's speed and direction and making the presence known to other vessels through consistent and recognizable fog sound signals.