Cartesian equation of vectors dermine if lines are parallel

Question

Determine whether the following planes and lines are parallel. If they are parallel, show whether the line lies in the plane.

(i) $\mathbf r = (3, 1, 2) + t(1, -1, 2)$ and $3x +y - z = 8$

How do you do these type of questions?

Answer 1

computing the dot product we get $$[3;1;-1]\cdot[1;-1;2]=3-1-2=0$$ this means that the straight line and the plane is parallel

Answer 2

Hint:

We see immediately that the point of the line $(3,1,2)$ (the point for $t=0$) is on the plane. Now verifies that also the point for $t=1$ is on the plane.