Am I able to just put the planes point into the equation and leave it at that? or am I wrong here?
My parametric equation:
(x,y,z) = (3,-2,1)+ t(2,1,-3) + s(1,-2,4)
2026-03-29 05:11:03.1774761063
Is this parametric equation correct?
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Incorrect; that parametric plane does not include the point $(1,-2,4)$!
But you can use $(x,y,z) = s(3,-2,1)+ t(2,1,-3) + (1-s)(1,-2,4)=s(2,0,-3)+ t(2,1,-3) + (1,-2,4)$
Now it is easy to see that for s=t=0 the plane includes $(1,-2,4)$, and for $s=1$ you get points over the line.