Lines in space
Problem 9.390 · medium
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 2 t - 1, 3 t + 4, 2 t - 4 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 1, 3 s + 1, 2 - 2 s \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}-12\\4\\6\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ 0 \](P₂ − P₁)·(d₁ × d₂): zero means coplanar, so the lines meet.✓ Proved
- \[ \left[\begin{matrix}1\\7\\-2\end{matrix}\right] \]At t = 1, s = 2 both lines pass through <1, 7, -2>.✓ Proved
Answer \( \text{intersecting}\text{ at } (1, 7, -2) \)
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 |
| 3 | ✓ 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 intersect based on them being coplanar. Coplanarity (checked in step 2) is a necessary but not sufficient condition for intersection; the lines could be parallel and distinct. The solution fails to check if the direction vectors are parallel (step 1 computes a non-zero cross product, implying they are not parallel, but the logic in step 2 ignores this distinction). While the final point is correct, the reasoning that 'coplanar implies meet' is mathematically false for general lines.
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 intersect based on them being coplanar. Coplanarity (checked in step 2) is a necessary but not sufficient condition for intersection; the lines could be parallel and distinct. The solution fails to check if the direction vectors are parallel (step 1 computes a non-zero cross product, implying they are not parallel, but the logic in step 2 ignores this distinction). While the final point is correct, the reasoning that 'coplanar implies meet' is mathematically false for general lines.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly concludes the lines intersect based on them being coplanar. Coplanar lines can be parallel and distinct; one must verify that the parameters t and s actually yield the same point, which is done in step 3, but step 2's logic ('zero means coplanar, so the lines meet') is mathematically false.gpt-oss:20b: pass 2026-10-08
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.