Inside a warehouse a conveyor cable follows the line (x - 1)/3 = (y + 2)/0 = (z - 4)/(-2)…

Step 1. The meaning of the zero. The equation says $y + 2 = 0 \times \lambda$, so $y = -2$ at every point. The technician is right. The cable lies in the vertical plane $y = -2$.

Step 2. The direction cosines. The direction ratios are $(3, 0, -2)$ and $\sqrt{9 + 0 + 4} = \sqrt{13}$. In the direction of increasing $x$ they are $l = \dfrac{3}{\sqrt{13}} \approx 0.832$, $m = 0$ and $n = -\dfrac{2}{\sqrt{13}} \approx -0.555$.

Step 3. The floor. The points are $x = 1 + 3\lambda$, $y = -2$, $z = 4 - 2\lambda$. Then $z = 0$ gives $\lambda = 2$ and $x = 7$. The cable meets the floor at $(7, -2, 0)$.

Step 4. The distance. From $(1, -2, 4)$ to $(7, -2, 0)$: $\sqrt{36 + 0 + 16} = \sqrt{52} = 2\sqrt{13} \approx 7.211$ metres.