Two survey markers stand at the points A(2, -3, 1) and B(8, 1, -3), with coordinates in metres…

Step 1. The direction ratios. $(8 - 2, 1 + 3, -3 - 1) = (6, 4, -4)$.

Step 2. The length. $AB = \sqrt{36 + 16 + 16} = \sqrt{68} \approx 8.246$ metres.

Step 3. The direction cosine $n$. $n = \dfrac{-4}{\sqrt{68}} \approx -0.485$.

Step 4. The line. Use the simpler direction $(3, 2, -2)$. Then $x = 2 + 3\lambda$, $y = -3 + 2\lambda$ and $z = 1 - 2\lambda$.

Step 5. The ground. $z = 0$ gives $\lambda = \dfrac{1}{2}$. So $x = 3.5$ and $y = -3 + 1 = -2$.