После ужина мы поедем домой на машине.

Because после requires the genitive case. The noun ужин changes to ужина in the genitive singular.

So:

• ужин = dinner
• после ужина = after dinner

This is a very common pattern:

• после работы = after work
• после урока = after the lesson
• после фильма = after the film

It is the genitive singular of ужин.

The base form is:

• ужин = dinner

After после, it becomes:

• после ужина = after dinner

For many masculine nouns, the genitive singular often ends in -а or -я.

Поедем is the 1st person plural future form of поехать, which means to set off / to go by vehicle.

So мы поедем means:

• we will go
• more literally: we will set off and go (by some form of transport)

This fits well with на машине, because the sentence is about going somewhere by car.

Both are possible in Russian, but they mean slightly different things.

• мы поедем домой = we will go home / we will set off for home
• мы будем ехать домой = we will be going home / we will be on the way home

In this sentence, поедем is more natural because it focuses on the future action as a whole: after dinner, we’ll go home.

Будем ехать would emphasize the process or duration of traveling, not just the fact that the trip will happen.

Yes. It is related to ехать.

A useful basic distinction is:

• ехать = to go by transport in one direction / at one time
• ездить = to go by transport repeatedly, habitually, or in various directions

The perfective partner of ехать is usually поехать.

So:

• ехать → imperfective
• поехать → perfective
• мы поедем = we will go / we will set off

Because домой means to home / homeward — it shows motion toward home.

Compare:

• дома = at home
• домой = home, to home

So:

• Мы дома. = We are at home.
• Мы едем домой. = We are going home.

This is a very important distinction in Russian.

Because домой already contains the idea of movement toward home. Russian treats it as a special adverb meaning homeward / to home.

So Russian says:

• идти домой = to go home
• ехать домой = to go home by transport

not something like to the home in a literal word-for-word way.

In Russian, means of transport are often expressed with на + prepositional case:

• на машине = by car
• на автобусе = by bus
• на поезде = by train
• на метро = by metro

So на машине is the normal way to say by car.

This is a very common question.

• на машине = by car, using a car as transport
• в машине = in the car, inside the car

Examples:

• Мы поедем домой на машине. = We’ll go home by car.
• Мы сидим в машине. = We are sitting in the car.

So in your sentence, на машине is correct because it tells you how you are going home.

Because after на in this meaning, Russian uses the prepositional case.

The base form is:

• машина = car

After на, it becomes:

• на машине = by car

This is a regular feminine noun pattern:

• книга → в книге
• школа → в школе
• машина → на машине

Yes. Russian word order is fairly flexible, although some orders sound more neutral than others.

Your sentence:

• После ужина мы поедем домой на машине.

is natural and neutral.

You could also say:

• Мы поедем домой на машине после ужина.

That also makes sense, but it places после ужина later, so the time expression is less prominent.

Putting После ужина first is very natural because it sets the scene: After dinner, ...

Yes, often you can.

Russian verb endings usually show who the subject is. Поедем already means we will go, so:

• После ужина поедем домой на машине.

can also be correct, especially in conversation.

Adding мы makes the subject more explicit. It can sound slightly clearer or more emphatic.

It suggests going by some means of transport, not on foot.

Russian often distinguishes:

• идти / пойти = go on foot
• ехать / поехать = go by vehicle

So in this sentence:

• поедем
• на машине

match each other very naturally.

The sentence is specifically about traveling home by car, not walking home.

Yes. Russian usually does not have articles like a or the, so после ужина can mean:

• after dinner
• after the dinner

The exact meaning depends on context. In most everyday situations, English would simply translate it as after dinner.

It sounds like a one-time future event.

That is partly because of поедем, which is a perfective future form. It presents the action as a single complete event in the future:

• after dinner, we’ll go home by car

If you wanted to talk about a repeated habit, Russian would more likely use different wording, often involving ездить or some adverb showing repetition.

The main stresses are:

• После
• у́жина
• мы
• пое́дем
• домо́й
• на маши́не

So a learner-friendly version is:

После у́жина мы пое́дем домо́й на маши́не.

Stress matters a lot in Russian pronunciation, so it is good to learn words together with their stress.