"The answer is in the positive" vs "the answer is positive" vs "the answer is ‘yes’"

They all mean roughly the same, though there are some differences. Let's say I asked the question:

Are you going to the party?

In this case you could use all three to indicate you are going to the party.

Let's say I asked instead:

Are you going to the party or not going to the party?

In casual conversation if you said "the answer is yes", then people would certainly understand that you were going, but it isn't strictly an answer to the question: the answer is one of the choices, and some might pedantically think that saying "yes" means you are either going or not going, which is to say, the answer is always yes, since everyone is either "going" or "not going" to the party. As I say though, that would be considered quite pedantic.

Let me offer a different example where this is a bit clearer:

Did the coin flip come up heads or tails?

Here if you answered "yes" people would roll their eyes at you, because obviously every coin flip is either heads or tails. You'd have to say "tails", for example, to convey any information. The party sentence has the same problem, except that one of the options is more "yes"-like than the other. So in casual conversation that would be more than sufficient.

However, if you say "the answer is in the positive" it means you are choosing the positive of the two options, namely that you are going to the party, and no grammar pendant could question your clarity.

So correct answers to "are you going to the party or not going to the party?" would be "I am going" or "I am not going", and you can use various substitutes instead of those literal words. For example, using an attribute of the phrase "in the positive", or using a shortened form such as "not" or "going".

The first two answers I really don't think there is much different in their meaning in this sort of context.