"Where is our director?"

Translation: Gdzie jest nasz dyrektor?

[locanisem]

Why do we not use the instrumental case here?

[immery]

because "nasz dyrektor" =our director is a subject of the sentence, and "gdzie" works like adverb.

[locanisem]

Indeed, thank you!

[roman_f1]

"Gdzie jest nasz szef"? - why not?

[Jellei]

"Szef" = "Boss", "director" doesn't have to be your boss in fact.

[alfaro.sal]

Is dyrektor considered as a masc personal noun?

[Jellei]

Yes. Although it doesn't matter in this sentence.