Lines in space
Problem 9.546 · easy
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 4 - 3 t, 3 t + 1, - t - 4 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 2 - 3 s, 2 s + 3, 3 s - 7 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}11\\12\\3\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ -7 \](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 despite finding they are coplanar (cross product non-zero and scalar triple product zero). Coplanar non-parallel lines must intersect, so the correct classification is intersecting, not skew.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly concludes the lines are skew despite finding they are coplanar (cross product non-zero and scalar triple product zero). 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 calculates the cross product of the direction vectors as non-zero, implying the lines are not parallel, but then uses a flawed logic for intersection. More critically, the comment on line 2 claims that (P2-P1) dot (d1 x d2) being zero means the lines meet; this condition only implies they are coplanar, not that they intersect. Furthermore, the actual cross product of < -3, 3, -1 > and < -3, 2, 3 > is < 11, 12, 3 >, which is non-zero, so they are not parallel. However, checking for intersection: 4-3t = 2-3s => 2 = 3(t-s). 3t+1 = 2s+3 => 3t-2s = 2. Solving these gives t=1, s=1/3? No. 3(t-s)=2 => t-s=2/3. 3t-2s=2. Multiply first by 2: 6t-6s=4. Subtract: (3t-2s)-(6t-6s) = 2-4 => -3t+4s = -2. This is getting messy. Let's just check the provided logic. The logic 'zero means coplanar, so the lines meet' is false. Coplanar lines can be parallel or intersecting. If they are not parallel and coplanar, they intersect. If they are not parallel and not coplanar, they are skew. The calculation in line 2 says the scalar triple product is -7, which is NOT zero. Therefore, the lines are NOT coplanar. Since they are not parallel (cross product is non-zero) and not coplanar (scalar triple product is non-zero), they are skew. The conclusion 'skew' is correct, but the reasoning in the comments is completely wrong. Line 2 says '-7 = -7' and comments 'zero means coplanar'. The value is -7, not zero. So the lines are not coplanar. The comment implies that because the value is -7 (which is not zero), they don't meet? Or does it imply that since it's not zero, they are skew? The comment is misleading/wrong because it defines the condition for meeting as 'zero', but then presents a non-zero value. A student would be confused about whether -7 means they meet or not. Actually, if the comment says 'zero means coplanar, so the lines meet', and the value is -7, the implication is that they do NOT meet. But the phrasing is terrible. It states a condition for meeting, then shows a value that fails that condition. It doesn't explicitly conclude 'therefore skew'. It just shows the calculation. However, the prompt asks to review the sentences. The sentence in line 2 is '(P2-P1) dot (d1 x d2): zero means coplanar, so the lines meet.' This is a false statement. Coplanar lines do not necessarily meet (they could be parallel). Also, the calculation result is -7, so they are not coplanar. The final answer is skew. The solution is technically correct in its final answer but the explanatory text is mathematically false ('coplanar ... so the lines meet'). This is a 'misleading' or 'error' severity. I will call it error because the statement 'coplanar implies meet' is false.gpt-oss:20b: fail (error) 2026-10-12 — The solution incorrectly states that (P₂−P₁)·(d₁×d₂)=0, implying coplanarity and intersection. In fact the dot product equals –7≠0, so the lines are skew. The conclusion that they intersect is false.
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.