De arts schreef pillen voor die ik moet innemen.

Because voorschrijven is a separable verb in Dutch.

• Full verb: voorschrijven = to prescribe

• In a main clause, the conjugated part goes in the normal verb position, and the prefix moves to the end:

• De arts schreef pillen voor. = The doctor prescribed pills.

So in your sentence:

• schreef = the past tense verb
• voor = the separated prefix belonging to voorschrijven

This is very common in Dutch with separable verbs, such as:

• opbellen → Ik bel je op.
• aandoen → Hij doet het licht aan.

Because in a normal main clause, Dutch usually puts the finite verb in the second position.

So the basic structure is:

• De arts = subject
• schreef = finite verb in second position
• pillen = object
• voor = separated prefix at the end

That gives:

• De arts schreef pillen voor

If you used voorschreef as one word in a main clause, that would sound wrong in standard Dutch, because separable verbs normally split there.

Die refers to pillen.

So:

• pillen = pills
• die ik moet innemen = that I have to take

This is a relative clause.

Dutch uses die for:

• de-words in the singular
• all plurals

Since pillen is plural, die is the correct relative pronoun.

Compare:

• de man die daar staat = the man who is standing there
• het boek dat ik lees = the book that I am reading
• de pillen die ik moet innemen = the pills that I have to take

Because pillen is plural.

A useful rule:

• die = for de-words and plural nouns
• dat = for het-words in the singular

So:

• de pil → de pil die...
• de pillen → de pillen die...
• het medicijn → het medicijn dat...

That is why die is correct in your sentence.

Because in modern standard Dutch, when you have a subordinate clause with a modal verb plus another verb, the usual order is:

• subject + modal + infinitive

So:

• die ik moet innemen

Here:

• ik = subject
• moet = modal verb
• innemen = infinitive

You may sometimes see different verb orders in longer clauses, but for learners, ik moet innemen is the normal and safest pattern.

Good question. Innemen is also a separable verb:

• innemen = to take, ingest, take in

In a main clause, it can split:

• Ik neem de pillen in.

But after a modal verb like moeten, the second verb stays as an infinitive at the end:

• Ik moet de pillen innemen.

So in your sentence:

• die ik moet innemen

the verb stays together because it is in the infinitive.

Because after a modal verb such as moeten, Dutch uses the bare infinitive, not te + infinitive.

So you say:

• ik moet innemen
• ik kan komen
• ik wil slapen
• ik mag gaan

Not:

• ik moet te innemen ❌

This is similar to English, where we say I must take rather than I must to take.

Moet is the singular present tense of moeten.

In this sentence, it means must / have to.

Forms of moeten:

• ik moet
• jij moet
• hij/zij/u moet
• wij moeten
• jullie moeten
• zij moeten

So:

• ik moet innemen = I have to take (them)

Because the object is already understood from pillen.

The relative clause die ik moet innemen literally means:

• that I must take

English often also leaves out them in a relative clause:

• the pills that I have to take

You could think of the object as already built into the relative structure, so Dutch does not need an extra pronoun there.

They are close, but not always identical.

• arts = physician, medical doctor; a more formal or professional term
• dokter = doctor; very common in everyday speech

In many contexts they can both refer to the same person, but arts often sounds a bit more formal or official.

So:

• De arts schreef pillen voor sounds natural and slightly formal.
• De dokter schreef pillen voor is also very natural in everyday Dutch.

Yes. That is also correct.

It uses the perfect tense:

• heeft voorgeschreven = has prescribed / prescribed

Compare:

• De arts schreef pillen voor...
  simple past; often used in narration or past context

• De arts heeft pillen voorgeschreven...
  perfect tense; very common in spoken Dutch when talking about a past action relevant now

Both are grammatical. The choice depends on style and context.

Both are possible, but they are not exactly the same.

• pillen = pills, tablets
• medicijnen = medicines, medication

If the doctor specifically prescribed tablets, pillen is fine.
If you want a broader word, medicijnen works too.

For example:

• De arts schreef medicijnen voor die ik moet innemen.

That sounds natural as well, though innemen especially fits medicine you swallow or ingest.