Find the shortest distance between the lines (x − 1)/2 = (y + 1)/1 = (z − 2)/(−2) and (x − 4)/1 = y/2 = (z − 1)/2.

1. 1 · Write what is given and what is to be found. Given: l₁ through (1, −1, 2) with DRs 2, 1, −2, and l₂ through (4, 0, 1) with DRs 1, 2, 2. To find: the shortest distance d.
2. 2 · Draw the figure or sketch. Draw two non-meeting lines and the common perpendicular PQ with right angles marked. Write PQ = d.
3. 3 · Write the formula. d = |(b⃗₁ × b⃗₂) · (a⃗₂ − a⃗₁)| / |b⃗₁ × b⃗₂|
4. 4 · Write the vectors.
   • a⃗₁ = î − ĵ + 2k̂, a⃗₂ = 4î + 0ĵ + k̂, so a⃗₂ − a⃗₁ = 3î + ĵ − k̂
   • b⃗₁ = 2î + ĵ − 2k̂, b⃗₂ = î + 2ĵ + 2k̂
5. 5 · Find the cross product.
   • b⃗₁ × b⃗₂ = (1 × 2 − (−2) × 2)î − (2 × 2 − (−2) × 1)ĵ + (2 × 2 − 1 × 1)k̂
   • = 6î − 6ĵ + 3k̂
   • |b⃗₁ × b⃗₂| = √(36 + 36 + 9) = √81 = 9
6. 6 · Find the dot product. (6)(3) + (−6)(1) + (3)(−1) = 18 − 6 − 3 = 9
7. 7 · Check and box the answer. d = |9|/9 = 1 unit. Since d ≠ 0, the lines are skew.

Scoring tip: Write the formula before substituting. If the cross product goes wrong, the formula line still earns a mark.