Why is look used as a transitive verb in the phrase look you in the eye?
I checked look in Cambridge Dictionaries and found only an intransitive look, not a transitive one.
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I don't understand why you guys think this question so complex. There is no need to label a word transitive or intransitive if anyone can clearly understand what a sentence means without ambiguousness; hence, it means that it is also grammatically correct.
Let's look over the questioned sentence again : “look you in the eye”.
Can you figure out by reading it that there is another way to interpret the sentence? Definitely not, and it is understandable enough because there is no ambiguousness in it.
What I mean is that if a sentence is fixed to be interpreted in one way, it is both grammatically correct and acceptable, needless to think of whether a verb is used transitively or intransitively.
Although it may appear to be, look in to look you in the eye is not transitive, nor is it the word that behaves oddly in the sentence. Rather, it is the you that behaves oddly.
In Old English there were four or five grammatical cases--depending on how you reckon them up--for which (most) nouns and adjectives inflected: nominative, accusative, genitive, dative, and instrumental. The dative and instrumental cases were already conflating by the time of the earliest written records, however, and before long the accusative case was likening itself to the dative, such that, come Middle English, there were only nominative, accusative, and genitive inflections--but the accusative case continued to be used in a dative manner.
In Modern English, we only represent grammatical case in our pronouns: I, we, he, she, they are subjective (nominative), me, us, him, her, them objective (accusative and dative), and my, our, his, her, their possessive (genitive). Apart from those, we tend to do the job of grammatical case either by a word's position in a sentence or by a preposition: i.e. by the sword (preposition + noun) is the new way to say þon sweorde (noun in the instrumental case).
Nevertheless, there are still some stragglers from the Old English cases, which, despite having been stripped of their inflections, still behave the same. One such straggler is the indirect object of ditransitive verbs:
I gave him it.
I gave it to him.
In the former, him is a straggler from the Old English dative case: it denotes the recipient of something. In the latter, to him serves this dative function, despite being accomplished by means of a preposition instead; it is, effectively, the 'new way' to say it. As I have said in my comments above, woe is me (woe is to me; I am the recipient of woe) is another such example.
So, to look you in the eye may be made to comply with the 'new way' by giving it a preposition:
To look to you in the eye.
But this sounds off, which is because the dative case was sometimes used where we would expect the genitive: i.e. it was sometimes used to denote possession. If we keep this in mind, we may recast the sentence as follows:
To look in your eye(s).
Which not only sounds natural but also represents the exact meaning of the phrase to look you in the eye.
In the comments of the original post some people have been comparing to look (you) in the eye to to look daggers at. I want to begin by saying that I do not know how the latter idiom ever came to be, but I can assure you that the two did not come about by way of the same process. If I had to guess, I would say that the daggers in to look daggers at were an adverbial genitive: i.e. a noun in the genitive case used as an adverb. Similar constructions include: I work nights, I work days, I always eat breakfast, I sometimes eat breakfast, I will do it anyways, etc.
Note that the adverbial genitive is going out of fashion, and as such some adverbial genitives are being re-analyzed as plural nouns used adverbially: hence adverbs like sometime in let's do that sometime.
Look was used as a transitive verb with the same definition many years ago, but it is now obsolete in that manner. Now it is intransitive, needing a preposition in order to show what exactly you are looking at. However, in modern informal speech, look is used transitively very often (although that doesn't mean it's grammatically correct).
Informal speech is usually spoken quick. You want to get the point across in an easy to understand, short and snappy manner. "I looked him in the eye" is short and sweet in informal speech, compared to "I looked at him in the eye."
To me all the examples have a slightly formal, or perhaps old fashioned color to them. But they are all perfectly valid and understandable.
"Look something in the eyes" is an idiom. Probably comes from an older transitive definition of "look".
Etymonline says it comes from Olde English and lists the origin of some idioms with "look" but not "look X in the eyes" but anyway that's a better start than what I had here originally.