What is the correct way to say "where are you going?"?
Also, can someone please explain that の in "あなたはどこに行くの"?
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Both ways are correct, but there are lots of ways of saying "where are you going" depending on politeness levels etc. You can also replace the
に particle with the
へ particle in many instances, and in fact you can go without both
へ in many informal situations. Often
あなたは isn't necessary as it can be determined by context.
In Japanese there isn't any distinction between "where are you going", "where are you going to go", "where do you go", "where will you go", "where will you be going" etc, and it's usually determined by context which is meant.
This is by no means a complete list and the explanations may need to be corrected, but I hope it will give some idea of the different ways of saying it:
どこ行くの more informal, omitting
どこに行くのかな informal but "softer", "where are you going I wonder?"
どこに行くのかしら feminine version of the above
どこに行くのだろう not strictly male, but I think it's used more by males similar to the above
どこに行くのでしょうか a formal version of
どこに行くのですか slightly more polite
どこに行くつもりですか／どこに行く予定ですか polite, "where do you intend to go"
どこ行きますか a polite but colloquial form
どこに行きますか polite form
Both works, although it'd be better if
あなたは was elided (unless there is a need to specifically refer to the person, otherwise it can have either an accusatory feel or it can have the effect of appearing intimate with the person).
に can also be elided.
の at the end behaves something like a sentence softener. It tends to have a feminine undertone. It is related to the sentence ending pattern
~のだ which roughly means "It is that ~". The question is derived from
The are both correct. It's just two different expression of politeness.
The first is polite (丁寧語). It is the form that usually ends with ます です.
The latter is less polite and can be used between friends. It uses the form that usually exists in dictionary to end sentences. (What should be form be called? In Japanese it may be called くだけた言い方)
And there is a note on the above page:
which says that it is usually used in plain text (that does not use ます です).