More than the result (that I already have printed in my book), I'd be interested in the procedure to switch from a non parametric to a parametric equation of a plane in the Euclidean space. Here is the exercise:
Find parametric equations describing the plane $$x_1 - 2x_2 +3x_3 = 6.$$
Thanks.
Since a surface is 'intrinsically' 2D, you're going to need 2 variables in any parametrization of the plane --- you can also notice that using a single variable gives equations for curves, and we want to 'smear' the curves in another direction to get a surface, hence the second variable coming into play.
The easiest way to parametrize is if you can write one variable in terms of the other two; in our case, we can write $x_1 = 6+2x_2 -3x_3$. Therefore, using the parameters $s$ and $t$ for our parametrization, we can write $$x_1 = 6+2s -3t,$$ $$x_2 = s,$$ $$x_3 = t.$$
