Lines in space
Problem 9.386 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle - 2 t - 4, 2 t, 3 - 3 t \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle - 4 s - 3, 4 s - 1, 2 - 6 s \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}0\\0\\0\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- The directions are parallel and P₂ is not on L₁, so the lines are parallel and distinct.
Answer \( \text{parallel} \)
Lines: 1 proved, 1 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 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 parallel and distinct, but they are actually identical (coincident). Point P2(-3, -1, 2) lies on L1 (at t=-1/2), so the lines are not distinct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution claims the lines are parallel and distinct, but they are actually identical (coincident). Point P2(-3, -1, 2) lies on L1 (at t=-1/2), so the lines are not distinct.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution claims the lines are parallel and distinct, but the direction vectors are scalar multiples (d2 = 2*d1) and the lines actually intersect at t=1, s=1/2 (point <-6, 2, 0>). Thus, they are not distinct parallel lines; they are the same line (coincident). The check for P2 on L1 was performed incorrectly or omitted.gpt-oss:20b: fail (misleading) 2026-10-08 — The solution states that P₂ is not on L₁ without providing any verification. A student would learn that the conclusion follows automatically from the parallelism, which is not justified; the intersection test should be shown.
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.