A straight pipe is to run above a flat roof without touching it…

Step 1. The condition. $\vec{b} \cdot \vec{n} = 2 \times 1 + 2k - 4 = 2k - 2$. It is $0$ when $k = 1$.

Step 2. The distance. The pipe is parallel to the roof, so the distance from the point $(1, 0, 1)$ to the plane is the distance of the whole pipe. The value is $2 + 0 - 1 - 3 = -2$. The length of the normal is $3$. The distance is $\dfrac{2}{3} \approx 0.667$ metres.

Step 3. The plane through the pipe. A plane $2x + 2y - z = c$ is parallel to the roof, and it contains the pipe if it contains the point $(1, 0, 1)$. Then $c = 2 + 0 - 1 = 1$. The direction $(1, 1, 4)$ is perpendicular to the normal, so the whole pipe lies in that plane.