I know one of possible correct answers for the question 'Do you want to see her?' is – Yes, I do.
But is also grammatically correct an answer: – Yes, I want. ?
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The rule of thumb: If the question begins "Do you..." then the answer is "I do." If the question begins "Will you..." then the answer is "I will."
If you are so nervous at your wedding that you cannot pay attention to the ritual questions you are asked, then just remembering the first two words of the question will get you through without too much trouble.
In the US, we do not answer "I want". But I believe elsewhere you may sometimes hear "I want", say in Ireland? Even the the US you could answer "I want to."
The convention is that you always use the first verb of the sequence. In a simple two-verb situation the first verb is often a modal verb so it may be said that you use the modal verb but the rule actually applies to any sequence of verbs.
Would you have been going to have started painting by Wednesday? Yes I would.
The verb do might be thought to be a special case as we use do in questions when we don't in positive statements. So you could answer
Do you want to eat?
I want to eat.
I want to.
But it appears that do sticks to the rules when you just use pronoun and verb.
It is disputed how much the Celtic languages have influenced English. It is my belief that the influence is significant, particularly in relation to really basic grammar and there is a specific feature of those languages which means their speakers are particularly used to this problem: they do not have words for "yes" and "no" so they use the verb, in positive or negative form, albeit without pronoun, to mean "yes" or "no". And just like in English they always use the first one. Thus there is a long history of this problem and its solution in Britain and Ireland.
Is she laughing? Is/isn't.