Lines in space
Problem 9.548 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 2 t, 3 t - 1, -3 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle - 3 s - 2, 2, - 2 s - 1 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}-6\\4\\9\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ 42 \](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 comment for line 2 claims that a zero scalar triple product implies the lines meet (intersect). This is false; it only implies the lines are coplanar. Coplanar lines can be parallel and distinct, in which case they do not intersect. The solution fails to distinguish between coplanar and intersecting.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The comment for line 2 claims that a zero scalar triple product implies the lines meet (intersect). This is false; it only implies the lines are coplanar. Coplanar lines can be parallel and distinct, in which case they do not intersect. The solution fails to distinguish between coplanar and intersecting.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The comments on both lines are factually incorrect. A non-zero cross product implies the lines are not parallel, and a non-zero scalar triple product implies the lines are not coplanar (and thus do not intersect). The solution correctly calculates that the lines are skew, but the reasoning provided in the text is the exact opposite of the truth.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.