Lines in space
Problem 9.228 · medium
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle t - 1, 3, 2 t \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 2 s, 3, 3 s + 1 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}0\\1\\0\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ 0 \](P₂ − P₁)·(d₁ × d₂): zero means coplanar, so the lines meet.✓ Proved
- \[ \left[\begin{matrix}-2\\3\\-2\end{matrix}\right] \]At t = -1, s = -1 both lines pass through <-2, 3, -2>.✓ Proved
Answer \( \text{intersecting}\text{ at } (-2, 3, -2) \)
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: fail (error) — The cross product of the direction vectors is incorrectly computed; it should be <0,3,0> instead of <0,1,0>. This makes the first sentence misleading, though the final intersection point is correct.qwen3.6:27b-mlx: fail (error) — The solution incorrectly concludes the lines intersect because they are coplanar. Coplanarity is a necessary but not sufficient condition for intersection; one must also verify that the specific parameter values yielding the intersection point are consistent across both lines (which they are here, but the logic in step 2 is flawed as it implies coplanarity alone guarantees intersection).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly concludes the lines intersect because they are coplanar. Coplanarity is a necessary but not sufficient condition for intersection; one must also verify that the specific parameter values yielding the intersection point are consistent across both lines (which they are here, but the logic in step 2 is flawed as it implies coplanarity alone guarantees intersection).gpt-oss:20b: fail (error) 2026-10-04 — The cross product of the direction vectors is incorrectly computed; it should be <0,3,0> instead of <0,1,0>. This makes the first sentence misleading, though the final intersection point is correct.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly concludes the lines are intersecting because they are coplanar. Coplanarity is a necessary but not sufficient condition for intersection; the lines are actually parallel and distinct (since the point on line 1 is not on line 2), so they do not intersect.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.