Lines in space
Problem 9.462 · medium
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle t, t + 3, 2 t - 2 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 2 s + 6, 1 - 2 s, s + 4 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}5\\3\\-4\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ 0 \](P₂ − P₁)·(d₁ × d₂): zero means coplanar, so the lines meet.✓ Proved
- \[ \left[\begin{matrix}2\\5\\2\end{matrix}\right] \]At t = 2, s = -2 both lines pass through <2, 5, 2>.✓ Proved
Answer \( \text{intersecting}\text{ at } (2, 5, 2) \)
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 |
| 3 | ✓ 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 intersect because they are coplanar. Coplanar lines can be parallel and distinct (non-intersecting). The cross product of direction vectors is non-zero, so they are not parallel, but the logic 'coplanar implies intersecting' is flawed without explicitly verifying a common point or showing non-parallelism first. While the final answer is correct, the reasoning in step 2 is mathematically invalid as a general test for intersection.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution incorrectly concludes the lines intersect because they are coplanar. Coplanar lines can be parallel and distinct (non-intersecting). The cross product of direction vectors is non-zero, so they are not parallel, but the logic 'coplanar implies intersecting' is flawed without explicitly verifying a common point or showing non-parallelism first. While the final answer is correct, the reasoning in step 2 is mathematically invalid as a general test for intersection.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution claims the lines are parallel because the cross product of direction vectors is zero, but the direction vectors <1, 1, 2> and <2, -2, 1> are not scalar multiples (their cross product is <5, 3, -4>, not zero). The logic that 'zero means coplanar, so the lines meet' is also flawed; non-parallel coplanar lines intersect, but parallel coplanar lines may be distinct. The final intersection point is correct, but the reasoning leading to it is mathematically invalid.gpt-oss:20b: fail (misleading) 2026-10-10 — The statement that a zero dot product of (P₂−P₁) with d₁×d₂ implies the lines meet is incorrect; it only shows coplanarity, not intersection. The conclusion should be that the lines are coplanar and then check for intersection separately.
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.