A power line passes through the point (1, -2, 0) with direction ratios (3, 4, 12), with lengths in…

Step 1. The cross product. $(7, 1, 12) \times (3, 4, 12) = (1 \times 12 - 12 \times 4,\ 12 \times 3 - 7 \times 12,\ 7 \times 4 - 1 \times 3) = (-36, -48, 25)$. Its length is $\sqrt{1296 + 2304 + 625} = \sqrt{4225} = 65$.

Step 2. The distance. $d = \dfrac{65}{13} = 5$ metres.

Step 3. The foot. $(7, 1, 12) \cdot (3, 4, 12) = 21 + 4 + 144 = 169$ and $\lvert \vec{b} \rvert^2 = 169$. So $t = 1$ and $F = (1 + 3, -2 + 4, 0 + 12) = (4, 2, 12)$.