Get any $3$ points in a plane from normal and one point

Question

I know how to get the normal of a plane from any 3 points, how can I do the inverse?

Suppose any $3$ points on the plane: $p_1$, $p_2$ ,$p_3$

We can get the vectors: $v_1 = p_2-p_1$,$v_2 =p_3-p_1$

And the plane normal n as: $n = v_1\times v_2$? If I got only $n$ and $p_1$, how can I get ANY other $2$ points: $p_2$ and $p_3$ ou two vectors on plane?

Looking for some vector form so I can easily write in matlab.

Answer 1

Expanding on @EmilioNovati 's answer in case you need more help. If we take vectors $\vec x, \vec n, \vec p$ where $\vec x = \left[ x_1,x_2\right],\vec n = \left[ n_1,n_2\right],\vec p = \left[ p_1,p_2\right]$ we know $$\vec n \cdot (\vec x - \vec p) = 0 \\ \vec n \cdot \left[x_1-p_1,x_2 - p_2\right] = 0 \\ n_1x_1-n_1p_1 + n_2x_2 - n_2p_2 = 0 \\ x_2 = \frac{n_1p_1+n_2p_2-n_1x_1}{n_2}$$ You know $\vec p$ and $\vec n$. You can substitute any $x_1$ to find the corresponding $x_2$

Answer 2

If you know the normal $\mathbf{n}$ and a point on the plane $\mathbf{p_1}$ than you know the equation of the plane: $$ \mathbf{n}\cdot(\mathbf{x}-\mathbf{p_1})=0 $$

so you can find all the points of the plane.