Defining Parametric function for a plane passing through points

Question

Question: Propose how to define parametrically with functions x(u,v), y(u,v), z(u,v) a plane passing through points with coordinates (-4,0,0), (0,4,0), (2,0,4).

My thought process would be to use a bilinear representation to define the middle points, but I have no idea if that is the correct first step using the formula P’ = P1 + u(P2 - P1)

May I have guidance on how should I tackle this question, I apologize if it seems trivial, I have no idea beyond trying using a bilinear representation, and even then I am not sure if it is the correct way of dealing with this.

Answer 1

Let $A=(-4,0,0), B=(0,4,0), C=(2,0,4).$

Taking arbitrarily point $A$ as the origin in the plane, let us express that the generic point $M=(x,y,z)$ is such that :

$$\vec{AM}=u \vec{AB}+v \vec{AC} \ \iff \ M=A+u \vec{AB}+v \vec{AC} \ \iff$$

$$\begin{cases} x&=&-4+4u+6v\\y&=& \ \ \ 0+4u+0v\\z&=& \ \ \ 0+0u+4v\end{cases}.$$

Verification : if $u=1$ and $v=0$, we find back $B$ ; if $u=0$ and $v=1$, we find back $C$.