Any one could it please spot the error I made in this question:

Q: A theme park has two zip lines. Sarah models both zip lines as straight lines using coordinates in meters. The ends of a wire are located at (0,0,0) and (0,100, -20). The ends of the other wire are at (10,0,20) and (-10,100, -5). Use Sarah's model to find the shortest distance between ziplines:

L1: r = s (100d-20k)

L2: (10i + 20k) + t (-20i + 100d – 25k)

Let P be a general point on line L1 and Q a general point on line L2:

Then: p = (100sj-20sk) and q = (10-20t) i + (100t) j + (20-25t) k

Vector PQ = q – p = ((10-20t) i + (100t) j + (20 – 25t) k) – (100sj – 20sk) = (10-20t) i + (100t – 100s) j + (20 – 25t + 20s) k

If the vector PQ is perpendicular to the two lines, then:

((10-20t) i + (100t – 100s) + (20 – 25t + 20s) k). (100d – 20k) = 0

10000t – 10000s – 400 + 500t – 40s = 0

10500t – 10040s = 400

((10-20t) i + (100t – 100s) + (20 – 25t + 20s) k). (-20i + 100d – 25k) = 0

-200 + 400t + 10000t – 10000s – 500 + 625t – 500s = 0

11025t – 10500s = 700

Is it correct? Simultaneous resolution gives t = 404/63 and s = 20/3. By linking this to the PQ vector and making it length, we get a response of 121.15 … when I'm pretty sure the answer is 50/3.