Which is correct?
1. What did you do for the winter vacation?
2. What did you do for winter vacation?
I think #2 is correct, if so ( or not ) how do I back this up with a specific rule?
#2 is the better of the two, yes.
http://grammar.ccc.commnet.edu/grammar/determiners/determiners.htm
Here is an excellent webpage from a college in Connecticut that explains the uses of articles in English.
Both sentences are grammatically correct, but they have slightly different meanings.
Sentence 1: "What did you do for the winter vacation?"
This sentence implies that there is a specific winter vacation that the speaker is referring to, possibly a specific holiday break or a particular winter season. The speaker wants to know what the listener did during that specific winter vacation.
Sentence 2: "What did you do for winter vacation?"
This sentence is more general and does not refer to a specific winter vacation. It can be interpreted as asking about any winter vacation that the listener may have had in the past or may have planned for the future.
To back this up with a specific rule, we can look at how we use prepositions in relation to time expressions. The preposition "for" typically indicates the duration or period of time for which an action takes place. In sentence 1, "for the winter vacation" suggests a specific period of time, making it more specific and definite. On the other hand, in sentence 2, the omission of the definite article "the" before "winter vacation" implies a more general or indefinite period. Therefore, both sentences are correct, but they convey different meanings based on the presence or absence of the definite article "the."