Lines in space
Problem 9.470 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 1, t - 2, 2 - 3 t \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 3, 2 s, 3 - 6 s \right\rangle \) 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: fail (misleading) — The solution states that point P₂ is not on line L₁, but no justification or calculation is provided to support this claim. Without verifying that the second line’s point does not satisfy the first line’s parametric equations, the conclusion that the lines are distinct parallel lines is unsupported.qwen3.6:27b-mlx: fail (error) — The solution claims P₂ is not on L₁ without verification. In fact, P₂(3,0,3) lies on L₁ (at t=2), so the lines are identical, not distinct.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-10 — The solution states that point P₂ is not on line L₁, but no justification or calculation is provided to support this claim. Without verifying that the second line’s point does not satisfy the first line’s parametric equations, the conclusion that the lines are distinct parallel lines is unsupported.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution claims P₂ is not on L₁ without verification. In fact, P₂(3,0,3) lies on L₁ (at t=2), so the lines are identical, not distinct.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution claims the lines are parallel based on a zero cross product, but the direction vectors <0, 1, -3> and <0, 2, -6> are indeed parallel. However, the solution fails to verify if the lines are distinct or coincident. Checking if P2(3,0,3) is on L1: 1=1, t-2=0 => t=2, 2-3(2)=-4 != 3. So they are distinct parallel lines. The logic in sentence 2 is correct, but the first line is marked 'unchecked' and the cross product calculation is not shown, just the result. Wait, the prompt says 'do not re-check the algebra of a line marked proved'. Line 1 is marked 'unchecked'. This is a contradiction in the prompt instructions or the line status. Let's look closer. Line 1 is `[equation, unchecked]`. The prompt says 'Each line... is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. It seems Line 1 is an equation that was NOT checked by CAS? Or is it a sentence? It's labeled equation. If it's unchecked, I should check it. d1 = <0, 1, -3>, d2 = <0, 2, -6>. d1 x d2 = <1*(-6) - (-3)*2, -3*0 - 0*(-6), 0*2 - 1*0> = <-6+6, 0, 0> = <0,0,0>. The equation is correct. The conclusion is correct. Why would it fail? Ah, look at the problem statement again. r1(t) = <1, t-2, 2-3t>. r2(s) = <3, 2s, 3-6s>. Directions are <0,1,-3> and <0,2,-6>. They are parallel. P1(1,-2,2). P2(3,0,3). Vector P1P2 = <2, 2, 1>. Is P1P2 parallel to d1? <2,2,1> vs <0,1,-3>. No. So they are distinct parallel lines. The solution says 'parallel'. This is correct. Is there a subtle error? 'The directions are parallel and P2 is not on L1, so the lines are parallel and distinct.' This is a correct statement. The verdict should be pass. Wait, let me re-read the prompt constraints. 'Reply with JSON only'. 'severity... error... misleading... style'. If it's correct, 'pass' with empty notes. Let me double check the cross product. d1 = <0, 1, -3>. d2 = <0, 2, -6>. Cross product is zero. They are parallel. Are they the same line? If they were the same line, they would intersect at infinitely many points. The question asks 'parallel, intersecting, or skew'. Usually 'parallel' implies distinct parallel lines in this context, or 'coincident' is a subset of parallel. If they are distinct, they are parallel. If they are coincident, they are also parallel (and intersecting). The standard classification is: Parallel (distinct), Intersecting (at one point), Skew. Coincident lines are often grouped with Parallel or Intersecting depending on the text. However, since P2 is not on L1, they are distinct. So 'parallel' is the correct answer. The solution is sound. Why did I think it might fail? Maybe the 'unchecked' tag? The prompt says 'do not re-check the algebra of a line marked proved'. Line 1 is 'unchecked'. This implies I SHOULD check it? Or does it mean the CAS didn't check it? If I am supposed to check it, and it is correct, then it passes. If the tag 'unchecked' is a mistake in the solution format provided to me, I should ignore it if the math is right. The math is right. The sentence is right. Verdict: pass.gpt-oss:20b: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.