Both arise and rise are irregular verbs.
When an opportunity, problem, or new state of affairs arises, it begins to exist. This is the most common meaning of arise.
He promised to help Rufus if the occasion arose.
A serious problem has arisen.
The feeling of hope arose from the people.
When something rises, it moves upwards.
The sun rises in the morning.
Clouds of birds rose from the tree-tops.
When someone who is sitting rises, they stand up. You can also use rise to say that someone gets out of bed in the morning.
Cambridge Dictionary also lists rise for "to begin to oppose or fight a bad government or ruler":
The people rose (up) against the oppressor/tyrant/dictator.
Please visit here to see other meanings listed for rise which all are connected with moving upward.
See Here.
Historically, mist was categorized as a meteorological or atmospheric condition. Hence, Webster's Dictionary of Synonyms (1942) lists mist in a category with haze, fog, smog, and brume:
Haze, mist, fog, smog, and brume agree in denoting an atmospheric condition which deprives the air near the earth of its transparency. ... Mist applies to a condition where water is held in suspension in fine particles in the air, floating or slowly falling in minute drops.
Figuratively, as Merriam-Webster's Eleventh Collegiate Dictionary (2003) observes, mist can also refer to various things suggestive of atmospheric mist:
4 a : a cloud of small particles or objects suggestive of a mist b : a suspension of a finely divided liquid in a gas c : a fine spray
In contrast, though vapor can be used to describe mistlike conditions, it has a stronger connection to gaseous rather than particulate solid or liquid states. It comes from the Latin word vapor meaning steam. Again from the Eleventh Collegiate, here are the first three definitions:
vapor n (14c) 1 : diffused matter (as smoke or fog) suspended floating in the air and impairing its transparency 2 a : a substance in the gaseous state as distinguished from the liquid or solid state b : a substance (as gasoline, alcohol, mercury, or benzoin) vaporized for industrial, therapeutic, or military uses; also : a mixture (as the explosive mixture in an internal combustion engine) of such vapor with air
Given the association of vapor with heating or sublimating into lighter than air gases (and leaving aside the stubborn fact that vapor has long been used as a way to describe fogs and—inevitably—mists), one might argue that a vapor in many instances rises, while a mist tends to descend slowly.
Incidentally, Noah Webster, in his Compendious Dictionary of the English Language (1806) gives only the heated or sublimated sense of vapor:
Vapor n. a fluid rendered volatile and elastic by heating, fume
Samuel Johnson, Dictionary of the English Language (1756) has a more varied (and interesting) take on the word:
VAPOUR. s. [vapor, Lat.] 1. Any thing exhalable ; any thing that mingles with the air. Milton. 2. Wind, flatulence. Bacon. 3. Fume ; steam. Newton. 4. Mental fume ; vain imagination. Hammond. 5. Diseases caused by flatulence, or by diseased nerves ; melancholy ; spleen. Addison.
Best Answer
The three have similar properties but have different size scales.
Haze is large scale like smog that you can see over long distances like obscuring a skyline or the horizon. I associate it with warmer dryer weather. It is very vague.
Fog is scientifically a low lying cloud, but when you're inside it, things are just not as bright and you can't see as far depending on how thick it is, and things feel a little cool and humid at the same time. There's very little differentiation inside, it's just heterogeneously a little less bright, and you can't see as far and it looks whiter in the distance, but there are no noticeable cloudy areas. Outside of it like from a mountain top you really do see that it is a cloud.
Mist is to me a kind of rain, or barely rain at all just a noticeable wetness in the air that an umbrella just really doesn't do anything to stop. You don't really see it falling but standing out in it, you eventually get wet.
Also, since language isn't logically consistent, a 'mister' outputs mist which looks like a small scale fog, a fog machine outputs very small scale fog that is hardly at all wet.
Hazy, foggy, and misty tend to follow all these but not always perfectly and usage would be modified as needed by idioms.