A straight cable runs along the line (x - 1)/2 = (y - 2)/3 = (z - 3)/4, with coordinates in metres…

Step 1. The vectors. $\overrightarrow{AP} = (-1, -2, -2)$ and $\vec{b} = (2, 3, 4)$.

Step 2. A normal. $\overrightarrow{AP} \times \vec{b} = ((-2) \times 4 - (-2) \times 3,\ (-2) \times 2 - (-1) \times 4,\ (-1) \times 3 - (-2) \times 2) = (-2, 0, 1)$.

Step 3. The plane. Use $(2, 0, -1)$, which is $-1$ times the normal. The constant from $P$ is $2 \times 0 + 0 - 1 = -1$. So $2x - z = -1$ and $k = -1$.

Step 4. The check. The point $A$ gives $2 - 3 = -1$. The direction $(2, 3, 4)$ gives $4 - 4 = 0$, so the whole line lies in the plate.

Step 5. The distance. $\dfrac{1}{\sqrt{4 + 1}} = \dfrac{1}{\sqrt{5}} \approx 0.447$ metres.

Step 6. The $x$-axis. Put $y = z = 0$: $2x = -1$, so $x = -0.5$.