It's an adverb. He went in that direction.
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AlpheccaStarsWhy are you calling "that way" a determiner phrase?https://en.wikipedia.org/wiki/Determiner_phrase
Persian Learnerhttps://en.wikipedia.org/wiki/Determiner_phraseWe don't typically get into these controversial topics in this grammar section of the forum. It might be better to post this in our Linguistics forum, though even there, there might not be a lot of member
Persian LearnerSo, could determiner phrases function as adverbials?Yes, but 'that way' is not called a determiner phrase except in some theoretical works in linguistics. The mainstream term for 'that way' is noun phrase (NP).