1. Are you planning to go to Australia for your vacation?

2. Are you planning to go to Australia on your vacation?
3. Are you planning to go to Australia for the purpose of your vacation?
4. Are you planning to go to Australia at the time of your
vacation?

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Does #1 mean #2 or #3? Does #1 mean #4? Is there any difference between #1 and #2?

They all mean approximately the same thing, but 3 and 4 are very wordy!

1 and 2 are fine and are used a lot.

1. Yes, #1 means the same thing as #2. Both sentences are asking if the person is intending to go to Australia during their vacation.

2. #3 is slightly different from #1 and #2. This sentence is asking if the person is going to Australia specifically for the purpose of their vacation, implying that they might have another reason to visit Australia besides vacationing.
3. #4 is similar to #1 and #2 but it includes the phrase "at the time of your vacation." This sentence is asking if the person has plans to be in Australia specifically during the time of their vacation, suggesting that they might have plans for other times as well.
Overall, there are slight differences in the nuances of each sentence, but they are quite similar in general meaning.

1. "Are you planning to go to Australia for your vacation?" asks whether you have made plans to travel to Australia during the time you have designated for your vacation. This question specifically asks about the destination of your vacation.

2. "Are you planning to go to Australia on your vacation?" has the same meaning as the first question. The preposition "on" is used interchangeably with "for" in this context, so there is no significant difference between #1 and #2.

3. "Are you planning to go to Australia for the purpose of your vacation?" asks if your intent in going to Australia is specifically for the purpose of your vacation. This question seeks to understand if Australia is your chosen destination for leisure or relaxation.

4. "Are you planning to go to Australia at the time of your vacation?" asks whether you have plans to visit Australia at the same time as your scheduled vacation. This question focuses on the timing of your travel rather than the purpose or destination.