Lines in space
Problem 9.388 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle - 2 t, - 2 t - 4, 1 - 3 t \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 3 - s, 2 s - 3, 4 - 3 s \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}12\\-3\\-6\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ 15 \](P₂ − P₁)·(d₁ × d₂): zero means coplanar, so the lines meet.✓ Proved
Answer \( \text{skew} \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the closest approach of the two lines, found numerically |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly concludes the lines are skew despite calculating that they are coplanar (the scalar triple product is zero). Since the direction vectors are not parallel and the lines are coplanar, they must intersect, not be skew.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly concludes the lines are skew despite calculating that they are coplanar (the scalar triple product is zero). Since the direction vectors are not parallel and the lines are coplanar, they must intersect, not be skew.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly concludes the lines are skew. The cross product of the direction vectors is non-zero, so they are not parallel. The scalar triple product is zero, meaning the lines are coplanar. Since they are coplanar and not parallel, they must intersect, not be skew.gpt-oss:20b: fail (misleading) 2026-10-08 — The second sentence incorrectly states that a zero dot product implies the lines meet; it only implies coplanarity, not intersection. The solution also omits the final conclusion that the lines are skew.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/lines_in_space, checked 2026-10-08 with SymPy 1.14.0.