Lines in space
Problem 9.304 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle t + 4, 2 t, -3 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle s + 6, 2 s - 3, -3 \right\rangle \) 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.Reviewed
Answer \( \text{parallel} \)
Lines: 1 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the direction vectors as parallel via the cross product and implicitly verifies they are distinct (since the lines do not intersect, as confirmed by the constant z-coordinate -3 and different x-intercepts for the same parameter value, or simply by checking if a point from one line lies on the other). The conclusion is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the direction vectors as parallel via the cross product and implicitly verifies they are distinct (since the lines do not intersect, as confirmed by the constant z-coordinate -3 and different x-intercepts for the same parameter value, or simply by checking if a point from one line lies on the other). The conclusion is correct.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution claims the lines are parallel, but the direction vectors <1, 2, 0> and <1, 2, 0> are identical, meaning the lines are parallel. However, checking if P2(6, -3, -3) is on L1: t+4=6 => t=2; 2t=-4 != -3. So they are distinct parallel lines. Wait, the cross product of <1,2,0> and <1,2,0> is indeed <0,0,0>. The logic holds. Let me re-read the prompt. 'Stated answer: parallel'. The solution says 'parallel and distinct'. This is correct. Why did I think it was wrong? Ah, I need to check if the cross product calculation was actually performed or just asserted. Line 1 is an equation marked 'unchecked' but the prompt says 'computer algebra system has already checked... do not re-check the algebra of a line marked proved'. Line 1 is marked 'unchecked' in the text provided? No, it says '[equation, unchecked]'. Wait, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is an equation. The prompt says 'do not re-check the algebra of a line marked proved'. Line 1 is NOT marked proved. It is marked 'unchecked'. This implies I should check it? Or does 'unchecked' mean the CAS didn't check it? The prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Line 1 is NOT marked proved. It is marked 'unchecked'. This is ambiguous. Usually, in these tasks, if it's not marked proved, I should check it. Let's check the cross product. d1 = <1, 2, 0>. d2 = <1, 2, 0>. Cross product is <0, 0, 0>. The equation is correct. The sentence says 'P2 is not on L1'. P2 is (6, -3, -3). L1 is (t+4, 2t, -3). t+4=6 -> t=2. 2(2)=4 != -3. So P2 is not on L1. The lines are parallel and distinct. The answer 'parallel' is correct. Is there a subtle error? 'Parallel' usually implies distinct in some contexts, or just same direction. If they were the same line, they would be 'coincident' or 'identical'. The problem asks 'parallel, intersecting, or skew'. If they are coincident, they intersect at infinitely many points. Since they are distinct, they are parallel. The solution is correct. Why would it fail? Maybe the cross product check is insufficient? No, for lines in 3D, parallel directions + distinct points = parallel lines. If directions are parallel and points are same, they are the same line. The solution correctly identifies they are distinct. I will mark it pass.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.