Lines in space
Problem 9.547 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle t + 1, t - 2, - 3 t - 4 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 3 s + 1, 2 s, -7 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}6\\-9\\-1\end{matrix}\right] \]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. A reviewer's objection to the explanation stands; see 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: fail (misleading) — The second sentence incorrectly concludes that the lines meet because the dot product is non‑zero; in fact a non‑zero value means the lines are not coplanar, so they are skew.qwen3.6:27b-mlx: fail (error) — The solution incorrectly concludes the lines are skew. The calculations show the direction vectors are not parallel and the lines are coplanar, which implies they intersect. The stated answer contradicts the derived mathematical facts.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-12 — The second sentence incorrectly concludes that the lines meet because the dot product is non‑zero; in fact a non‑zero value means the lines are not coplanar, so they are skew.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly concludes the lines are skew. The calculations show the direction vectors are not parallel and the lines are coplanar, which implies they intersect. The stated answer contradicts the derived mathematical facts.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly concludes the lines are skew. The calculations show the cross product of direction vectors is non-zero (not parallel) and the scalar triple product is zero (coplanar), which implies the lines intersect. The stated answer 'skew' contradicts the derived result.gpt-oss:20b: fail (error) 2026-10-12 — The solution incorrectly concludes that the lines meet. The dot product (P₂−P₁)·(d₁×d₂) equals –15, not zero, so the lines are not coplanar and therefore are skew. The final statement should reflect that, not that they intersect.
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 2026-10-12 with SymPy 1.14.0.