Lines in space
Problem 9.541 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle - 2 t - 2, 2 t - 2, 0 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 3 s + 1, 2 s - 3, - s \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}-2\\-2\\-10\end{matrix}\right] \]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: fail (error) — The solution incorrectly states that (P₂−P₁)·(d₁×d₂)=−4 equals zero, concluding coplanarity and intersection. In fact −4≠0, so the lines are not coplanar and are skew, matching the correct answer.qwen3.6:27b-mlx: fail (error) — The solution incorrectly concludes the lines are skew despite finding they are coplanar (scalar triple product is zero) and not parallel. Coplanar, non-parallel lines must intersect, so the correct classification is intersecting, not skew.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-12 — The solution incorrectly states that (P₂−P₁)·(d₁×d₂)=−4 equals zero, concluding coplanarity and intersection. In fact −4≠0, so the lines are not coplanar and are skew, matching the correct answer.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly concludes the lines are skew despite finding they are coplanar (scalar triple product is zero) and not parallel. Coplanar, non-parallel lines must intersect, so the correct classification is intersecting, not skew.qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly concludes the lines are skew based on a flawed logical structure. Step 1 calculates the cross product of direction vectors, which is non-zero, correctly implying the lines are not parallel. However, Step 2 calculates the scalar triple product (volume of parallelepiped) as -4, which is non-zero, implying the lines are NOT coplanar (and thus skew). The comment in Step 2 claims 'zero means coplanar, so the lines meet,' but the calculation shows it is NOT zero. Furthermore, even if they were coplanar, they might not intersect (they could be parallel, but we already ruled that out). The final answer 'skew' is actually correct for these specific lines, but the reasoning in the comments is contradictory and confusing: Step 2's comment suggests that if the value were zero, they would meet, but since it is -4, they don't. The bigger issue is that the solution doesn't explicitly state the condition for skew lines (non-parallel and non-coplanar). It presents two calculations but the text annotations are misleading or incomplete. Specifically, the annotation for step 2 says 'zero means coplanar, so the lines meet'. This is false; coplanar lines can be parallel (no intersection) or intersecting. Since we established they are not parallel, coplanar would imply intersection. But the value is -4, so they are not coplanar, hence skew. The verdict is 'skew', which is correct. Why is it a fail? The prompt asks to review the solution. The solution provides the correct final answer. However, the annotations are technically incorrect or misleading. Step 2 annotation: 'zero means coplanar, so the lines meet'. This is a non-sequitur. Coplanar does not automatically mean they meet (they could be parallel). Although we know they aren't parallel from step 1, the statement 'so the lines meet' is not a direct consequence of 'coplanar' alone without the 'not parallel' premise. More importantly, the solution doesn't explicitly conclude 'skew' from the non-zero result in step 2. It just shows the calculation. The stated answer is 'skew'. The steps support 'skew' (not parallel, not coplanar). Is there an error? Let's re-read carefully. Step 1: d1 x d2 != 0 -> Not parallel. Step 2: (P2-P1) . (d1 x d2) != 0 -> Not coplanar. Not parallel + Not coplanar = Skew. The logic holds. The annotation in step 2 is slightly sloppy ('so the lines meet' ignores the parallel case, but in context of step 1, it's okay). Wait, look at the equation in step 2. It says `-4 = -4`. This is a tautology, not a calculation result. It doesn't show the calculation `(P2-P1) . (d1 x d2) = -4`. It just asserts `-4 = -4`. This is a 'style' or 'misleading' issue because it doesn't show the work. But is it an 'error'? The prompt says 'equation... checked by CAS'. If the CAS checked `-4 = -4`, that's trivial. The real issue is that the solution doesn't actually *calculate* the scalar triple product in the equation line, it just states the result is -4 (implicitly) by showing `-4 = -4`? No, it shows `-4 = -4`. This is not a derivation. It's just a true statement. It doesn't link the problem data to the number -4. A student would not learn how to get -4. This is 'misleading' or 'style'. However, usually 'error' is reserved for wrong math. The math conclusion is correct. The annotations are the problem. The annotation in step 2 says 'zero means coplanar, so the lines meet'. This is logically flawed because coplanar lines can be parallel. Since step 1 proves they are not parallel, the implication holds, but the statement itself is false in general. This teaches a student something false: 'Coplanar lines always intersect'. That is a 'misleading' severity. Also, the solution doesn't explicitly state the final conclusion 'skew' in the text, it just relies on the 'Stated answer'. The steps don't lead to a sentence 'Therefore, the lines are skew'. They just show two equations. This is incomplete. I will mark it as misleading because the annotation teaches a false general principle.gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.