I need to find a plane that goes through the points $A=(2,0,2)$ and $B=(4,1,0)$, that is parallel to the line?
$$r(t) = (0,3,-2) + t\langle1,-1,1\rangle$$
or if you want it in parametric equations:
$$x = t, \ y = 3 - t, \ z = t - 2.$$
How do I find a plane that goes through two points? and how do I decide if it is parallel to the line?
Notice that if the plane is parallel to L, then the vector normal to the plane is then perpendicular to the line. Find the/ (a) vector normal to the plane, then you have two points in the plane, and you're done.
And there are infinitely-many planes that go through any two given points; there are infinitely-many planes that even go through a given line. Once you're given a vector normal to the plane, and two points in the plane, you're done ( although, given two points, you can find N using their cross-product. )
If the line lies in the plane, you can translate the plane to avoid containing the line.