I am looking for a word which describes the act of doing something most people think is impossible or when you want to show to someone that you are capable of doing something maybe like revenge or negative encouragement. I'd be happy to see some words related to psychology too.
closed as unclear what you're asking by Edwin Ashworth, JJJ, Spagirl, John Lawler, curiousdannii Jun 23 '18 at 1:58
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If you're talking about doing something that people don't think is possible (or perhaps they just think it's not at all likely for you), then you are beating the odds:
to succeed despite not having a good chance of succeeding · Most new restaurants fail, but this one somehow managed to beat the odds.
But if you're talking about doing something they didn't think you had it in you to do, then you are acting out of character:
not in accord with a person's usual qualities or traits · his rudeness was completely out of character
Or, perhaps, if you're doing it only because they have repeatedly said you couldn't (or wouldn't) do it, then you are acting in spite of them:
in defiance or contempt of : without being prevented by · succeeded in spite of their opposition
(This is related to doing something out of spite if, like revenge, the intent is negative.)
Against the grain - doing something unique, where everyone else (every grain) is against you
Thinking outside the square - quite a rare trait for thinking with a different set of boundaries and limitations. Based on a classic IQ question you can find with your favourite search engine
To stick your neck out means to take a risk.
From Cambridge Dictionary:
I’m really sticking my neck out by investing my money in this idea.
To stick one's neck out "take a risk" is first recorded 1919, American English.