Lines in space
Problem 9.387 · easy
Find parametric and symmetric equations of the line through \( \displaystyle P(2, -5, 1) \) and \( \displaystyle Q(3, 1, -6) \).
- \[ \left[\begin{matrix}1\\6\\-7\end{matrix}\right] \]A direction vector.✓ Proved
- r(t) = P + t·(Q − P); solving each coordinate for t gives the symmetric form.Reviewed
- \[ \left[\begin{matrix}3\\1\\-6\end{matrix}\right] \]At t = 1 the line reaches Q.✓ Proved
Answer \( x = t + 2,\ y = 6 t - 5,\ z = 1 - 7 t;\quad \frac{x - (2)}{1} = \frac{y - (-5)}{6} = \frac{z - (1)}{-7} \)
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 the parametric equations. The sentence accurately describes the standard method for deriving symmetric equations from parametric ones, and the final result matches the stated answer.
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 the parametric equations. The sentence accurately describes the standard method for deriving symmetric equations from parametric ones, and the final result matches the stated answer.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly calculate the direction vector Q - P, which is the crucial step connecting the points to the final equations. It presents the direction vector as an unchecked fact without showing the subtraction [3-2, 1-(-5), -6-1], making the derivation incomplete and potentially confusing for a student.gpt-oss:20b: pass 2026-10-08
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.