Mold cleanup - "while I'm in there"

I have recently purchased an older home and have had to do some plumbing work. In order to access the upstairs plumbing, I've taken down the entire kitchen ceiling. It has exposed the plumbing in question and some evidence of minor water leaks - nothing recent. There are some small mold spots on the wood under the tub, but they look old and dead. My question is, would spraying the wood around these areas with white vinegar kill any mold that might still be living there? Nobody is living in the house yet and probably won't be for several weeks while I finish this work so the smell isn't a major issue. Also I'm considering one of the tea-tree oil products like Hygienic Aire or similar to place in the furnace return ducts and just allowing the fan to circulate the air for a couple of days. Are these good ideas or wastes of time/money?
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