all these three words have the meaning of "doing sth", but there are also difference:
1 meaning
"正" mainly about the time, "在" mainly about the status, and "正在" covers both time and status, like
---在干吗?---我在做作业呢。 ---what are you doing? ---i'm doing homework.
---在干吗?能来一下吗? ----我正做作业呢! ---what are you doing? can you come now? ---i'm doing my homework!
here, "正" emphsis on "i have sth to do at the moment now", of course u can use "在", but not so good as "正" here
2 grammar
when you use “正", you can not merely use "正+verb", while it is ok in "在" and ”正在“
---你在做什么?
---我在做作业。 ok
---我正在做作业。 ok
---我正做作业。 no----我正做作业呢! Ok --- you need to use "正+verb+呢/着"
3 在can not be followed by "从"(from), while 正 and 正在 is ok
他正从学校赶回来。OK
他正在从学校赶回来。OK
他在从学校赶回来。NO
4 在can be used to mean the actions are done again and again or lasts quite long, while 正 and 正在 can not.
正在做作业. doing homework now OK
一直正在做作业 NO
---一直在做作业。 ok
you can see when 正在and 在 can only be used in "doing sth now", but not "doing sth all the time"