Opvaskebørsten ligger ved vasken, og vaskekluden hænger over vandhanen.

Because Danish, like German, usually writes compound nouns as one word.

• opvaskebørste = opvask
  • børste
• vaskeklud = vaske
  • klud
• vandhane = vand
  • hane

English often uses separate words, such as dish brush or dish cloth, but Danish normally combines them into one noun. When a compound noun becomes definite, the definite ending is added only to the very end:

• opvaskebørste → opvaskebørsten
• vaskeklud → vaskekluden

Here -en is the singular definite ending, roughly equivalent to English the.

So:

• en opvaskebørste = a dish brush

• opvaskebørsten = the dish brush

• en vask = a sink

• vasken = the sink

• en vaskeklud = a dishcloth / washcloth

• vaskekluden = the dishcloth

• en vandhane = a tap / faucet
• vandhanen = the tap / faucet

English uses a separate word, the. Danish often puts that meaning onto the end of the noun.

Because Danish usually expresses definiteness by adding an ending to the noun itself.

So instead of:

• English: the sink
• Danish: vasken

This is very normal in Danish. A separate word such as den or det is mainly used when there is an adjective or some extra description:

• vasken = the sink
• den store vask = the big sink

So in your sentence, no separate the is needed.

That depends on the noun’s grammatical gender.

Danish has two genders:

• common gender → usually takes en in the indefinite form and -en in the definite form
• neuter gender → usually takes et in the indefinite form and -et in the definite form

In this sentence, all the nouns are en-words:

• en opvaskebørste
• en vask
• en vaskeklud
• en vandhane

So their definite forms all end in -en.

Unfortunately, you usually have to learn the gender with the noun, just like learning en vask rather than only vask.

Danish often prefers a position verb when describing where something is.

Common position verbs include:

• ligge = to lie / be lying
• stå = to stand / be standing
• sidde = to sit / be sitting
• hænge = to hang / be hanging

So:

• Opvaskebørsten ligger ved vasken
  literally: The dish brush lies by the sink
• vaskekluden hænger over vandhanen
  literally: the dishcloth hangs over the tap

This sounds more natural in Danish than using er everywhere. English often just says is, but Danish likes to specify the physical position.

Yes, that is grammatically possible, and a Danish speaker would understand it. But it is less specific and usually less natural than ligger.

Compare:

• er ved vasken = is by the sink
• ligger ved vasken = is lying / resting by the sink

Danish often sounds more idiomatic when you choose the position verb that matches the object’s physical state.

These are two different prepositional phrases.

• ved vasken = by / at / near the sink
• over vandhanen = over the tap / faucet

So ved gives a general location near something, while over describes a position above or draped over something.

In this sentence:

• the brush is located by the sink
• the cloth is hanging over the tap

So the prepositions are chosen to match the physical arrangement of the objects.

They are both in the present tense.

The infinitives are:

• at ligge = to lie
• at hænge = to hang

The present tense is formed with -r:

• ligge → ligger
• hænge → hænger

Because the infinitive already ends in -e, adding -r gives the spelling -er.

So:

• ligger = lies / is lying
• hænger = hangs / is hanging

Yes, it follows a very important Danish pattern: in a main clause, the finite verb is normally in the second position.

In the first clause:

• Opvaskebørsten = subject
• ligger = finite verb

In the second clause:

• vaskekluden = subject
• hænger = finite verb

So both clauses follow normal Danish main-clause word order.

You can also move another element to the front, but then the subject usually moves after the verb:

• Ved vasken ligger opvaskebørsten.

That is a classic Danish verb-second pattern.

Because og is linking two full clauses, each with its own subject and finite verb:

1. Opvaskebørsten ligger ved vasken
2. vaskekluden hænger over vandhanen

In written Danish, a comma before og in this kind of sentence is very common and standard. It helps show that two complete clauses are being joined.

So the comma is not random punctuation; it marks the boundary between two coordinated clauses.