Answer
(a) parallel
(b) perpendicular
(c) Line 1 is perpendicular to the plane, and Line 2 is parallel to the plane.
Work Step by Step
(a) Given the line is parallel to $\vec v$ and the plane has a normal vector $\vec n$, if the line is perpendicular to the plane, the two vectors $\vec v$ and $\vec n$ should be parallel to each other.
(b) If the line is parallel to the plane, the two vectors $\vec v$ and $\vec n$ should be perpendicular to each other.
(c) Given the plane equation as $x-y+4z=6$, we can identify the normal vector as $\vec n=\langle 1, -1, 4 \rangle$. From the line equations we can get: Line 1 $\vec v_1=\langle 2, -2, 8 \rangle$ and Line 2 $\vec v_2=\langle -2, 2, 1 \rangle$, we can see that $\vec v_1=2\vec n$ is parallel to $\vec n$ indicating that Line 1 is perpendicular to the plane, while $\vec v_2\cdot\vec n=-2-2+4=0$ indicating that Line 2 is parallel to the plane.
