Мы напрасно ждали такси у другого входа: машина стояла с противоположной стороны.

Ждали is the past tense plural form of ждать = to wait.

The subject is мы = we, so the verb must agree with it:

• я ждал / ждала = I waited
• ты ждал / ждала = you waited
• он ждал = he waited
• она ждала = she waited
• мы ждали = we waited
• они ждали = they waited

So Мы ждали simply means we waited.

Напрасно means in vain, for nothing, unsuccessfully, or needlessly.

So Мы напрасно ждали means We waited in vain.

It is very close to зря. In many situations they can be interchangeable:

• Мы напрасно ждали.
• Мы зря ждали.

Both mean roughly We waited for nothing / It turned out to be pointless to wait.

A useful nuance:

• зря is often more conversational
• напрасно can sound a bit more literary or slightly more formal

Такси is an indeclinable noun, which means its form usually does not change across cases.

So you get:

• такси приехало = the taxi arrived
• ждать такси = to wait for a taxi
• сесть в такси = to get into a taxi
• ехать на такси = to go by taxi

The word stays такси.

This is common with some borrowed words in Russian.

Because the verb ждать normally takes a direct object.

So:

• ждать автобус
• ждать поезд
• ждать такси
• ждать друга

This is different from English, where we say wait for something.
In Russian, you usually just wait something/someone grammatically, without a separate word for for.

Here у means by / near / at in the sense of location, and after у, Russian uses the genitive case.

So:

• вход = entrance
• другой вход = another / other entrance
• у другого входа = by the other entrance

Both words change to genitive:

• другой → другого
• вход → входа

This is a very common pattern:

• у дома = by the house
• у окна = by the window
• у станции = by the station

It can mean either, depending on context.

In this sentence, у другого входа most naturally means something like:

• at the other entrance
• at a different entrance

The idea is that we were waiting at the wrong entrance, while the taxi was elsewhere.

So другой here means other / different, not second in a counting sense.

The colon introduces an explanation of what came before.

Structure:

• Мы напрасно ждали такси у другого входа = We waited for the taxi in vain at the other entrance
• машина стояла с противоположной стороны = the car was on the opposite side

The second part explains why the waiting was in vain.

So the colon here works like:

• ..., because...
• ...; the reason was that...

It links the result and its explanation.

In Russian, it is very natural to avoid repeating the exact same word if the meaning is clear.

Here:

• такси = taxi
• машина = car

Since a taxi is a car, машина is a natural substitute. It sounds less repetitive.

So the sentence is effectively saying:

• We waited for the taxi at the wrong entrance: the car was standing on the opposite side.

This kind of variation is very normal in Russian.

Yes, стоять literally means to stand, but with vehicles it often means:

• to be parked
• to be standing there
• to be waiting there

So машина стояла does not mean the car was physically upright in some strange way. It simply means the car was there, stationary.

Very common examples:

• Машина стоит у дома. = The car is parked by the house.
• Автобус стоял у остановки. = The bus was standing at the stop.

Both patterns can exist in Russian, but they are used a little differently.

In this sentence, с противоположной стороны is a very natural expression meaning:

• on the opposite side
• from the opposite side

It is somewhat idiomatic and often used when describing where something is relative to where you are.

Breakdown:

• сторона = side
• с стороны = from the side / on the side of
• с противоположной стороны = from/on the opposite side

You may also hear на противоположной стороне, especially when talking about streets or physical sides in a more direct spatial way. But с противоположной стороны is perfectly natural here.

It is genitive singular.

That is because the preposition с here means something like from / on the side of, and in this use it takes the genitive.

So:

• противоположная сторона = the opposite side
• с противоположной стороны = from/on the opposite side

Changes:

• противоположная → противоположной
• сторона → стороны

Because both actions are presented as states/processes, not single completed events.

• ждали = were waiting / waited
• стояла = was standing / was parked

Ждать is naturally imperfective when you describe the process of waiting.
Стоять is also imperfective because it describes a state.

The sentence is basically painting a situation:

• we were waiting there
• the car was standing somewhere else

If you changed the aspect, the meaning would shift.

Russian word order is flexible, but the chosen order sounds natural and clear.

Current order:

• Мы напрасно ждали такси у другого входа
• машина стояла с противоположной стороны

This order emphasizes the story in a logical sequence:

1. what we did
2. where we did it
3. why it was pointless

You could rearrange some parts for emphasis, for example:

• Мы ждали такси напрасно у другого входа.
• У другого входа мы напрасно ждали такси.

These are possible, but the original sentence sounds smooth and neutral.

No, not here.

That is because другого is from другой = other / different.

It is not from друг = friend.

Compare:

• другой вход = another entrance
• вход друга = the friend’s entrance

So у другого входа definitely means by the other/different entrance.