I know that there are other questions on this site about this, but I haven't gotten from them what I want.
Say we have two points $a_1,a_2\in\Bbb R^2$. thus the line $A$ connecting them is given by $$A=\{a_1+t(a_2-a_1):t\in\Bbb R\}$$ Now lets say we have 3 nonlinear points $p_1,p_2,p_3\in\Bbb R^3$. What is the set-equation of the plane $P$ which connects these points?
The parametric equation of the plane passing through three not aligned points is for example
$$P=\{p_1+s\cdot (p_3-p_1)+t\cdot (p_2-p_1):s,t\in\Bbb R\}$$
Can you see why?