What is a cartesian equation for 3 space passing through 3 points?

Question

What does cartesian equation for 3 pace look like?

and is there any way to describe this equation using determinant?

Answer 1

If $\mathcal P$ is the plane $(ABC)$, then: $M \in \mathcal P \Leftrightarrow \det(\vec{AM},\vec{AB},\vec{AC})=0$