Equation of a plane from 2 lines

Question

I have two lines with the following equation

$$D1 : (x, y, z) = (2,0,0) + k(0,3,0)$$ $$D2 : (x, y, z) = (2,0,2) + k(0,0,1)$$

and I must find out the equation of the plane that they make. I made a quick visual representation of the two lines, and if I'm right they don't cross. There's one that goes straight vertical and the other one is parallel to the axis of Z.

Is an equation still possible or it's a trap issue?

Answer 1

The equation of any plane in $\mathbb{R}^3$ is given by $$\overrightarrow{\mathbf{n}}\cdot\overrightarrow{\mathbf{x}}=\overrightarrow{\mathbf{n}}\cdot\overrightarrow{\mathbf{p}}$$ where

1. $\overrightarrow{\mathbf{n}}$ is a vector orthogonal to the plane
2. $\overrightarrow{\mathbf{x}}=(x_1,\dotsc,x_n)$
3. $\overrightarrow{\mathbf{p}}$ is a point on the plane

The data you give is sufficient to construct $\overrightarrow{\mathbf{n}}$ and $\overrightarrow{\mathbf{p}}$. Indeed we may take $\overrightarrow{\mathbf{n}}=(0,3,0)\times(0,0,1)=(3,0,0)$ and $\overrightarrow{\mathbf{p}}=(2,0,0)$. Thus our equation becomes $3x=6$. The equation of our plane is then $ x=2 $.