Lines in space
Problem 9.550 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 1 - 2 t, t + 1, 2 t + 2 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle -1, - s - 1, 2 s + 1 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}4\\4\\2\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ -18 \](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 claims the lines are skew, but the calculations show the direction vectors are parallel and the lines are coplanar. Since they are parallel and coplanar, they must be either identical or disjoint parallel lines, not skew. The conclusion contradicts the derived facts.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution claims the lines are skew, but the calculations show the direction vectors are parallel and the lines are coplanar. Since they are parallel and coplanar, they must be either identical or disjoint parallel lines, not skew. The conclusion contradicts the derived facts.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution claims the lines are skew but calculates that the scalar triple product is zero, which implies the lines are coplanar. Since the direction vectors are not parallel, coplanar lines must intersect, contradicting the conclusion.gpt-oss:20b: pass 2026-10-12
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.