How to parameterise a plane? Ie. $2x+3y=z$

Question

I see in the text book, we can equate $x=\cos{}u$, $y=\sin{u}$, $z=v$, but seems to be for cylinders, how do we find a parametric representation of $2x+3y=z$?

Answer 1

Let $P:=\{(x,y,z)\in\mathbb{R}^3:2x+3y=z\}$.

$P=\{(x,y,z)\in\mathbb{R}^3:\exists t\in\mathbb{R}, \exists u\in\mathbb{R}, (x=t)\wedge(y=u)\wedge(z=2t+3u)\} =\{(x,y,z)\in\mathbb{R}^3:\exists t\in\mathbb{R}, \exists u\in\mathbb{R}, (x,y,z)=t(1,0,2)+u(0,1,3)\}=\mathbb{R}(1,0,2)+\mathbb{R}(0,1,3)$