Lines in space
Problem 9.301 · medium
Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle 3 t - 2, t - 4, 3 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle 1 - 2 s, - 3 s - 3, 3 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.
- \[ \left[\begin{matrix}0\\0\\-7\end{matrix}\right] \]d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
- \[ 0 \](P₂ − P₁)·(d₁ × d₂): zero means coplanar, so the lines meet.✓ Proved
- \[ \left[\begin{matrix}1\\-3\\3\end{matrix}\right] \]At t = 1, s = 0 both lines pass through <1, -3, 3>.✓ Proved
Answer \( \text{intersecting}\text{ at } (1, -3, 3) \)
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 claims the lines are parallel because the cross product of direction vectors is zero, but the cross product is actually <0, 0, -7>, which is non-zero. The lines are not parallel; they are intersecting.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution claims the lines are parallel because the cross product of direction vectors is zero, but the cross product is actually <0, 0, -7>, which is non-zero. The lines are not parallel; they are intersecting.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution claims the lines are intersecting, but the direction vectors <3, 1, 0> and <-2, -3, 0> are not parallel, and the cross product is <0, 0, -7>, not zero. The logic in step 2 is also flawed: a zero scalar triple product implies coplanarity, not necessarily intersection (they could be parallel and distinct). However, since they are coplanar and not parallel, they must intersect. The final point is correct, but the reasoning in steps 1 and 2 is mathematically incorrect regarding parallelism and the implication of the scalar triple product.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.