Two parallel rails of a gantry pass through A(1, 0, 0) and B(2, 2, 0), and both have direction…

Step 1. A normal. $\overrightarrow{AB} \times \vec{b} = (1, 2, 0) \times (1, 2, 2) = (2 \times 2 - 0 \times 2,\ 0 \times 1 - 1 \times 2,\ 1 \times 2 - 2 \times 1) = (4, -2, 0)$. Use $(2, -1, 0)$.

Step 2. The plate. The constant from $A$ is $2 \times 1 - 0 + 0 = 2$. So $2x - y = 2$ and $k = 2$. The point $B$ gives $4 - 2 = 2$.

Step 3. The distance from the origin. $\dfrac{2}{\sqrt{5}} \approx 0.894$ metres.

Step 4. The sensor. The value at $(3, -1, 5)$ is $6 + 1 - 2 = 5$. The distance is $\dfrac{5}{\sqrt{5}} = \sqrt{5} \approx 2.236$ metres.