Home›Calculus 3›Lines in space›Problem 9.542

Lines in space

Problem 9.542 · easy

Are the lines \( \displaystyle \mathbf r_1(t) = \left\langle t + 1, 3 - t, - 2 t - 3 \right\rangle \) and \( \displaystyle \mathbf r_2(s) = \left\langle s + 4, s + 5, 2 s - 5 \right\rangle \) parallel, intersecting, or skew? If they intersect, find the point.

\[ \left[\begin{matrix}0\\-4\\2\end{matrix}\right] \]
d₁ × d₂: zero exactly when the directions are parallel.✓ Proved
\[ -12 \]
(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: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly concludes the lines are skew. The calculation shows the cross product of direction vectors is non-zero (not parallel) and the scalar triple product is zero (coplanar), which implies the lines intersect. The stated answer 'skew' contradicts the derived result.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly concludes the lines are skew. The calculation shows the cross product of direction vectors is non-zero (not parallel) and the scalar triple product is zero (coplanar), which implies the lines intersect. The stated answer 'skew' contradicts the derived result.
qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly concludes the lines are skew. The calculation in step 2 shows the lines are coplanar (scalar triple product is zero), and since the direction vectors are not parallel (step 1), the lines must intersect. The stated answer 'skew' contradicts the derived result.
gpt-oss:20b: fail (error) 2026-10-12 — The solution incorrectly concludes that the lines are coplanar and therefore intersect, contradicting the stated answer that they are skew. The dot product test shows coplanarity, but the cross product is non‑zero, so the lines are not parallel and must intersect, not be 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 2026-10-12 with SymPy 1.14.0.