Why is it not possible to calculate planes in $\mathbb{P}^2$, when vectors in $\mathbb{P}^2$ represent vectors in $\mathbb{R}^3$ !?

Question

question: Why can't I calculate a plane in $\mathbb{P}^2$ with three given points?!

Explanation: I've read multiple times that planes in $\mathbb{P}^2$ are not defined. Only lines and points. But with the fact that each vector in $\mathbb{P}^2$ can represent vectors in $\mathbb{R}^3$, I do not understand why I cant calculate planes in $\mathbb{P}^2$. E.g. given the points $(3,4,1)$, $(3,4,2)$, $(2,4,2)$.

Answer 1

As $\mathbb P^2$ is having dimension $2$, a plane of $\mathbb P^2$ is just $\mathbb P^2$ itself.

Note: you may need to clarify further what you mean by to calculate.