To take...
For me, this one always uses "a".
To take a vacation.
However, if you change it to a noun (gerund) it can lose the "a":
Taking vacations is my favorite past-time.
I speak American English, so, it may be that "to take vacation" doesn't sound wrong to British English speakers.
To go on...
Both with and without "a" sound fine, but possibly with a small difference.
I went on a vacation last week. Where did you go?
I went on vacation last week. Oh, that explains why you weren't in the office.
To me, "to go on a vacation" seems more exciting than without a. I would use the second version, if I do not intend to tell people where I went, but rather, just want to tell them I was gone (although they might still ask anyway).
The perspective is a little different.
I went on a vacation to Italy. (the place you went to)
I went on vacation from work. (the place you left)
But you could also say,
I went on vacation to the Bahamas.
To be on...
Same as To go on..., the article just changes the emphasis a bit.
Hello, this is XYZ Management Services. How can I help you?
Could you transfer me to Susan?
I'm sorry, she's on vacation till next week, do you need help with your account?
The definite article is not used in this expression.
When we assign an entity membership in a class we use the indefinite article, regardless of how ‘determinate’ the entity is, because it is not the only member of the class.
For instance, we ordinarily say “I own a ’57 Chevy”—that is, “The car I own belongs to the class ’57 Chevy”. You own only one car, it is entirely determinate, you introduce it into the conversation with the definite article; but it is just one of many ’57 Chevys. You would only say “I own the ’57 Chevy” if you and your hearers were looking at a group of cars and you were identifying your car as the only ’57 Chevy in the group.
Likewise, when you speak of a particular individual known to your hearer, you use a definite determiner in identifying or naming her—my wife—but the indefinite article in assigning her to a class: “My wife is a graduate student in mediaeval English lit.”
In the same way, when you say “What a car!”, you are not identifying the car but asserting its membership in the class of “cars to which attention should be drawn”.
We never say "What the X," because that utterance is not intended to identify which X you are talking about. It does not enjoin the hearer to "Look at that X", it observes that that X is an X of a certain class: it is an X which excites your admiration and deserves your hearer's attention.
Best Answer
Usage of articles in English sometimes seems totally haphazard or random. Many times, it is just an idiomatic usage.
As a native speaker, I can say that the following are idiomatic in American English:
This is the case, even when we do not have a particular hospital in mind. And it doesn't matter whether there is one or more than one hospital that one can practically 'go to' (that is, is in the area). We use this for other typical locations, such as the library, the park, the grocery store, the mall, etc., even when there are many in a city or geographic area.
We also use the elevator even when there are five elevators one can take, and we do not have a particular elevator in mind. For example: 'Take the elevator (= any elevator) to the tenth floor' (not an elevator, although it would be grammatically correct).
The same for
This works the same as the hospital, as no particular doctor is in mind. It could be that a native speaker is conceiving of the doctor as a location (similar to the doctor's), but I'm not sure; and I'm a native speaker. I just use the language as other native speakers do. I didn't invent idiomatic expressions; I inherited them.
Nevertheless, grammatically it is okay to use a in either sentence above.
And the following would only be said with the indefinite article, unless a particular doctor is in mind:
And the idiomatic expression:
The use of the definite article for certain buildings (the hospital) or places (the park) even when there are more than one in a given geographic locale, such as a city or town, is shown in the following joke:
No particular hospital is being referenced, just any hospital (which one would think by logic that American English speakers would use only the indefinite article for, but this is not the case). And the answer to the joke is
If you stand in the middle of the road, you will, sooner or later, get hit by a car, and you will get injured, and you will wind up at the hospital.