How do I find the parametric representation of the $t_1\vec{u}+t_2\vec{v}+\vec{w}$ for a plane passing through these three points?

Question

The points this plane must go through are as follows: $(1,1,0), (-2,0,2),$ and $(2,1,1)$. I don't understand how to approach this problem.

Answer 1

The system of parametric equations of the plane is a linear combination of the points that belong to it, i.e. \begin{align} p(t_1,t_2) = t_1 \Bigg( \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} - \underbrace{ \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} }_{\text{reference}} \Bigg) + t_2 \Bigg( \begin{bmatrix} -2 \\ 0 \\ 2 \end{bmatrix} - \underbrace{ \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} }_{\text{reference}} \Bigg) + \underbrace{ \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix} }_{\text{reference}} \end{align}

Answer 2

Thed plane has the equation $$\vec{x}=[1;1;0]+\alpha[-3;-1,2]+\beta[1;0;1]$$ where $$\alpha,\beta$$ are the parameters.