Lines in space
Problem 9.381 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 1 - t, t + 3, -2 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle -2, 3 - s, - s - 1 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}-1\\-1\\1\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ 4 \](P₂ − P₁)·(d₁ × d₂): zero means coplanar, so the lines meet.✓ Proved
Answer \( \text{skew} \)
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 |
| 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 are skew based on a flawed logical chain. Step 1 shows the cross product of direction vectors is non-zero, correctly implying the lines are not parallel. Step 2 calculates the scalar triple product as 4 (non-zero), which correctly implies the lines are not coplanar. However, the comment in Step 2 claims 'zero means coplanar, so the lines meet,' which is false (coplanar lines can be parallel and distinct). More importantly, the final answer 'skew' is actually correct for this specific problem (since they are not parallel and not coplanar), but the reasoning provided in the comments is garbled and potentially misleading about the relationship between coplanarity and intersection. Wait, let me re-read the prompt's constraint. 'Check that... each conclusion follows from the lines before it'. The stated answer is 'skew'. The calculations show: 1. Not parallel. 2. Not coplanar. Therefore, they are skew. This logic is sound. Why did I think it was wrong? Ah, the comment in step 2 says 'zero means coplanar, so the lines meet'. This is a false statement in the comment. Coplanar lines do not necessarily meet (they could be parallel). Since the result is non-zero, the lines are not coplanar, hence skew. The conclusion 'skew' is correct. But the comment 'so the lines meet' is factually false as a general rule, and misleading in this context because it suggests that if the value *were* zero, they *would* meet, which isn't true (they could be parallel). The prompt asks to check if 'nothing would teach a student something false'. The comment teaches that coplanar implies intersecting, which is false. Thus, it is misleading/error in explanation.
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 are skew based on a flawed logical chain. Step 1 shows the cross product of direction vectors is non-zero, correctly implying the lines are not parallel. Step 2 calculates the scalar triple product as 4 (non-zero), which correctly implies the lines are not coplanar. However, the comment in Step 2 claims 'zero means coplanar, so the lines meet,' which is false (coplanar lines can be parallel and distinct). More importantly, the final answer 'skew' is actually correct for this specific problem (since they are not parallel and not coplanar), but the reasoning provided in the comments is garbled and potentially misleading about the relationship between coplanarity and intersection. Wait, let me re-read the prompt's constraint. 'Check that... each conclusion follows from the lines before it'. The stated answer is 'skew'. The calculations show: 1. Not parallel. 2. Not coplanar. Therefore, they are skew. This logic is sound. Why did I think it was wrong? Ah, the comment in step 2 says 'zero means coplanar, so the lines meet'. This is a false statement in the comment. Coplanar lines do not necessarily meet (they could be parallel). Since the result is non-zero, the lines are not coplanar, hence skew. The conclusion 'skew' is correct. But the comment 'so the lines meet' is factually false as a general rule, and misleading in this context because it suggests that if the value *were* zero, they *would* meet, which isn't true (they could be parallel). The prompt asks to check if 'nothing would teach a student something false'. The comment teaches that coplanar implies intersecting, which is false. Thus, it is misleading/error in explanation.qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer returned a non-objectgpt-oss:20b: fail (error) 2026-10-08 — The conclusion that the lines meet is incorrect: (P₂−P₁)·(d₁×d₂)=4≠0, so the lines are not coplanar and therefore skew.
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.