Dry and arid mean without moisture. Dry is the general word indicating absence of water (a dry well) or freedom from moisture (dry clothes). Arid has a very specific context of use. It suggests great or severe dryness in a region or climate, especially when it results in bareness or in barrenness: arid climate of Nevada; arid soil. If you wish to use it figuratively, remember it’s a harsher word than dry. We call something dry that is uninteresting or dull (a dry geography textbook). Arid suggests something devoid (barren) of interest, imagination, emotion, or vitality: an an arid performance by a burned out cast.
The verbs cross and traverse both refer to passing or extending over or across something. Saying hikers traverse or hikers cross a meadow to reach a road conveys the same information, although traverse is more formal. Note that the characteristic use of cross to mean “to intersect” or “pass over a line” cannot be replaced by traverse—that is, you can say that two paths crossed to indicate that they intersected, but not that they traversed. Traverse most often suggests passing over or covering a considerable distance, sometimes with difficulty. But it is also frequently used of physical artifacts such as pipelines, railways, roads, paths, and trails, all of which traverse regions and terrains.
Who will take the trophy in compete vs. vie? Compete is commonly used to talk about people participating in an organized competitive event (athletes competing in Tokyo for Olympic gold) or businesses trying to gain the same share of the market (cable channels competing for the 18-35 demographic). Vie frequently occurs outside these more structured contexts, as with vie for control or vie for power, or vie for a seat, a space, or a spot. Compete can almost always substitute for vie, but not vice versa, in part because vie is always followed by a preposition and an indirect object—vie for, vie with. When compete appears in one of these phrases, vie is almost always a solid alternative: The boys competed/vied for their father's attention.