Hi, I have a question, How to use at/in the back when I want to say, for example:
There's a bedroom in/at? the back of the house.
There's room for three people in/at? the back.
There's a parking lot in/at? the back of the mall?

I don't know when to use at/in...thanks.

In the above sentences taken out of context , I would use "in". But I think I would use "at" in those phrases if they were used to set up the background for further description or action.

Example: There's a parking lot in the back of the mall.
There's a parking lot at the back of the mall where you will find my blue truck.

"Example: There's a parking lot in the back of the mall.
There's a parking lot at the back of the mall where you will find my blue truck."

Oh man, this becomes a bit esoteric.

Compared to what?

What do you mean, compared to what?

I just wanted to say, in my very own clumsy way that those differences seem very subtle to me, almost non existing, but hey, it's not my native language.

Your example sounds like you can say "in the back of the mall". If so, I would disagree. You can only say "at the back of the mall ". Otherwise it sounds like something is located on the inside of the back which does not make sense.

I agree with Marianne10.

'At' denotes position only, 'in' means inside.

He was shot in the back of the head. (i.e. the bullet entered his head via the back)
The suitcase is in the back of the car. (i.e. in the boot (or 'trunk', if you prefer))

The car park is at the back of the supermarket. (i.e. behind the supermarket)
The suitcase is at the back of the cupboard. (i.e. located near the wall)

So for your examples:
There's a bedroom at the back of the house. (in might be arguable here, but at sounds more natural to me).
There's room for three people in/at the back.
* There's room for three people in the back of the car.
* There's room for three people at the back of the table.
There's a parking lot at the back of the mall. (If it was 'in' then parking would have to be provided actual building. But even then 'at' seems preferable to me.)

I think that with standalone "there is/are" phrases, the new info is after the "be" verb. Eg. "There is some milk." >"some milk" = new info.

If you start modifying the new info you are doing it in relation to presumably known info. Eg. "There is some milk {back of cookies}." {back of cookies}= known info.

So what is the most natural way to incorporate the known info {back of cookies}? Usually with known info you would use defining particles. So {back of cookies} becomes {the back of the cookies} But style requires a preposition.

"In" is much more natural with defined info than "at". Consider He is in school." vs "He is at school." You are more likely to use the former in reference to a school you know and with an interlocutor that also knows the school.

So "There is some milk {the back of the cookies}." becomes "There is some milk in the back of the cookies."

I think I got it...
So, I use 'in' when something is inside something else and 'at' for position.

That's a good rule of thumb, but in actual usage it is not really that simple (otherwise you probably would not have asked the question) .

You have to remember that "inside something" can be a conceptual container and not just a physical one. Same goes for position.

I think that the most valid explanations depend on discoursal factors such as new and known info.

Yeah I know....prepositions like in/at are very complicated, especially for beginners like me...I think the most important is to learn in a natural way, instead of focusing on grammar rules.