"Please look up difficult kanji in your dictionary."
"Hiku" is a Japanese word that happens to have a lot of idiomatic uses. It basically means something like the English verb "pull" or "draw" but it often seems simply to fill the role of verb in expressions such as "jisho wo hiku" or "piano wo hiku" where the action has, at least to the English speaking mind, litte to do with pulling. In such cases the idiom just has to be remembered and the appropriate translation has to be what describes the action required with the object of "hiku" as in "consult a dictionary" or "play a piano." This is more of a vocabulary problem than a grammar problem. It is similar to the problem of learning all the uses of "to fix" in English.
When I looked up the verb 引く（ひく）, it doesn't really mean "look up (something)." It may be more helpful to think of it as "to use (something)". If you think of it this way, the sentence means, Please use the dictionary for difficult kanji. And that means "the dictionary" is the direct object.
What confuses me, is why some of the sentences use the を participle and others use で for the dictionary.
In this case, the meaning of "ひく" is seaches for the meaning of the difficult kanji OR make sure the difficult kanji with dictionaries.
The "で" has been used for making sure it with something in this case. For example, "じしょ"で"ひく", this means to check about something it and then make sure it, and then "じしょ"を"ひく", this means to use dictionaries. BUT for example if you listen, "くるま"で"いく", this means to go somewhere by car. The meaning of "で" is changed like this.
Even though I am an native Japanese, it is difficult to understand it.
The particle "wa" marks the topic rather than the subject of the sentence. It says that the predication (the verbal structure to follow) applies to what it marks. Often the topic is the "subject" of the verb but it does not have to be. Here it is not. ("(What I'm talking about is) difficult kanji (and what I'm saying is) look them up in the dictionary.)
じしょをひく is a set phrase meaning "consult the dictionary." (ひく also means "pull" or "play an instrument," so maybe you're pulling the dictionary off the shelf?) If you're going to break up the phrase, I suppose you could use で to indicate the means by which you're doing something.
It is flexible but the direct object phrase almost always appears to the direct left of the verb. And the verb will almost always be at the end of a phrase. You can explore alternative word orders if you wish but native speakers are going to think you odd and foreign speakers might not understand what you're trying to say.
From what I understand, あなた is used only when you don't actually know the person's name. It's super formal and actually kind of rude in some situations. Generally, you'd either leave it out and let context do the work, or you'd actually say the person's name. The exception would be a wife talking to her husband, in which case あなた is kind of a pet name.
Posted by N. Mai
"学生達は辞書を引いて難しい漢字を調べます" May I ask why it's 引いて? How do the 2 clauses of this sentence connect?
I can't reply directly to this since we've hit maximum comment depth, so I'm posting this as a new comment instead.
引いて is simply the て-form of the verb 引く, and it is this form that is connecting the sentences. The て-form works by connecting different parts of a sentence: this can be used for a lot of different things, but they all have that in common.
E.g. in Duolingo's default translation for this sentence 「難しい漢字は辞書を引いてください。」, 引いて is being used to connect the sentence to ください and by so doing, making it into a polite request.
As for the specific usage of the て-form that you were asking about, allow me to answer that by reference to A Dictionary of Basic Japanese Grammar:
(A₁: the action or state expressed by the first predicate; A₂: the action or state expressed by the second predicate)
A₁ is the means by which someone does A₂ or the manner in which someone does A₂. Examples:
Boku wa aruite kaetta.
(Lit. I walked and went home. (=I went home on foot.))
Kenji wa isoide gohan o tabeta.
(Lit. Kenji hurried and ate his meal. (=Kenji ate his meal in a hurry.))
When we look back at our sentence, it can be easily broken down into two main parts: 「学生達は辞書を引きます。」(The students consult a dictionary) and 「難しい漢字を調べます」([They] look up difficult kanji). So if we break down the sentence as above:
The students consult a dictionary and look up difficult kanji. (=The students look up difficult kanji in a dictionary.)