Lines in space
Problem 9.382 · easy
Find parametric and symmetric equations of the line through \( \displaystyle P(0, -6, 0) \) and \( \displaystyle Q(-5, 3, -6) \).
- \[ \left[\begin{matrix}-5\\9\\-6\end{matrix}\right] \]A direction vector.✓ Proved
- r(t) = P + t·(Q − P); solving each coordinate for t gives the symmetric form.Reviewed
- \[ \left[\begin{matrix}-5\\3\\-6\end{matrix}\right] \]At t = 1 the line reaches Q.✓ Proved
Answer \( x = - 5 t,\ y = 9 t - 6,\ z = - 6 t;\quad \frac{x - (0)}{-5} = \frac{y - (-6)}{9} = \frac{z - (0)}{-6} \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 line passes through P at t = 0 and Q at t = 1 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the direction vector and applies the standard parametric and symmetric formulas. The logic is sound and the final answer matches the problem statement.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies the direction vector and applies the standard parametric and symmetric formulas. The logic is sound and the final answer matches the problem statement.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly calculate the direction vector Q - P, which is crucial for deriving the parametric equations. While the final answer is correct, the provided steps do not show the derivation of the vector <-5, 9, -6> from points P and Q, making the jump from the problem statement to the final equations unjustified and potentially confusing for a student.gpt-oss:20b: fail (error) 2026-10-08 — Sentence 3 incorrectly states the direction vector as Q instead of Q−P; the line’s parametric form is not properly derived from the correct direction vector (-5,9,-6).
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 2026-10-08 with SymPy 1.14.0.