A hook is fixed at the point P(1, 1, 1) in a workshop, with lengths in metres…

Step 1. The heights. The wire goes from height $0$ to height $2$ and passes the hook at height $1$. So the hook is the midpoint of the wire.

Step 2. The coordinates. The midpoint of $(a, 0, 0)$ and $(0, s, 2)$ is $\left(\dfrac{a}{2}, \dfrac{s}{2}, 1\right)$. It equals $(1, 1, 1)$, so $a = 2$ and $s = 2$.

Step 3. The check. The points $(2, 0, 0)$ and $(0, 2, 2)$ give the direction $(-2, 2, 2)$. The vector $P - (2, 0, 0) = (-1, 1, 1)$ is half of it, so the three points are in one line. The wire also lies in the plane $y - z = 0$ through $P$ and the $x$-axis, since $2 - 2 = 0$.

Step 4. The length. $\lvert (-2, 2, 2) \rvert = \sqrt{12} \approx 3.464$ metres.