رخصة القيادة مع السائق الآخر.

It can be understood as a full sentence in Arabic, especially in context.

Arabic often omits the present-tense verb to be. So where English says:

• The driver's license is with the other driver

Arabic can simply say:

• رخصة القيادة مع السائق الآخر

Grammatically, this is a nominal sentence:

• رخصة القيادة = the subject
• مع السائق الآخر = the predicate, literally with the other driver

So the missing is is completely normal.

This is an iḍāfa construction, often called a possessive or genitive construction.

Literally, it is:

• رخصة = license
• القيادة = the driving

Together:

• رخصة القيادة = the driving license / the driver's license

A few important things about iḍāfa:

• The first noun usually does not take ال
• The second noun is in the genitive
• The whole phrase can be definite or indefinite depending on the second noun

So رخصة القيادة is definite because القيادة is definite.

Because in an iḍāfa, the first noun normally does not take ال.

So Arabic says:

• رخصة القيادة

not:

• الرخصة القيادة

That second version is not correct for this meaning.

Even without ال on رخصة, the whole phrase is still definite because the second noun, القيادة, is definite. This is a very common feature of Arabic.

Because it is the second term of an iḍāfa.

In an iḍāfa:

• the first noun keeps its own role in the sentence
• the second noun is put in the genitive

So in full vowelled form, you could write:

• رُخْصَةُ القِيادَةِ

Here:

• رخصةُ is nominative because it is the subject of the sentence
• القيادةِ is genitive because it is the second part of the iḍāfa

مع usually means with.

In this sentence, it does not mean physical accompaniment only. It can also mean that something is in someone’s possession.

So here مع السائق الآخر means something like:

• with the other driver
• in the possession of the other driver

Also, مع is a preposition, so the noun after it must be in the genitive.

Because الآخر is an adjective here, and in Arabic adjectives normally come after the noun.

So:

• السائق = the driver
• الآخر = the other

Together:

• السائق الآخر = the other driver

Arabic adjectives follow the nouns they describe and agree with them in:

• definiteness
• gender
• number
• case

Since السائق is definite, the adjective must also be definite:

• السائق الآخر

not:

• السائق آخر
• الآخر السائق for this ordinary adjective use

Usually, yes, but the exact English translation depends on context.

الآخر can mean:

• the other
• another one in some contexts
• sometimes the latter in more formal writing

In this sentence, السائق الآخر most naturally means:

• the other driver

Yes.

Here is why:

• رخصة القيادة is definite as a whole because it is an iḍāfa whose second noun, القيادة, is definite
• السائق الآخر is definite because both the noun and adjective have ال

So the whole sentence refers to specific things:

• the driver’s license
• the other driver

In fully vocalized MSA, it would be:

• رُخْصَةُ القِيادَةِ مَعَ السّائِقِ الآخَرِ

Breakdown:

• رُخْصَةُ: nominative, because it is the subject
• القِيادَةِ: genitive, because it is the second noun in the iḍāfa
• مَعَ: preposition
• السّائِقِ: genitive, because it follows a preposition
• الآخَرِ: genitive, because it is an adjective describing السائق

In normal pause, many speakers will not pronounce all final short vowels clearly, but the grammatical endings are still there in formal analysis.

A helpful pronunciation is:

• rukhṣatu l-qiyādati maʿa s-sāʾiqi l-ākhari

Notes:

• رخصة = rukhṣa, with a strong kh
• القيادة = al-qiyāda
• مع = maʿa, with the letter ʿayn
• السائق = as-sāʾiq in connected pronunciation, because س is a sun letter, so ال assimilates
• الآخر = al-ākhar, with a clear kh

If you pause at the end in natural speech, you may hear something closer to:

• rukhṣat al-qiyāda maʿa s-sāʾiq al-ākhar

Yes, but the meaning becomes less definite.

Compare:

• رخصة القيادة = the driving license / the driver’s license
• رخصة قيادة = a driving license

So if you are referring to a specific license already known in context, رخصة القيادة is the better form.

If you just mean the general idea of a driving license, رخصة قيادة can work.