Lines in space
Problem 9.227 · easy
Find parametric and symmetric equations of the line through \( \displaystyle P(0, 4, 0) \) and \( \displaystyle Q(-2, -1, -1) \).
- \[ \left[\begin{matrix}-2\\-5\\-1\end{matrix}\right] \]A direction vector.✓ Proved
- r(t) = P + t·(Q − P); solving each coordinate for t gives the symmetric form.
- \[ \left[\begin{matrix}-2\\-1\\-1\end{matrix}\right] \]At t = 1 the line reaches Q.✓ Proved
Answer \( x = - 2 t,\ y = 4 - 5 t,\ z = - t;\quad \frac{x - (0)}{-2} = \frac{y - (4)}{-5} = \frac{z - (0)}{-1} \)
Lines: 2 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — Step 3 incorrectly states the direction vector as (-2,-1,-1) instead of the correct (-2,-5,-1), and thus incorrectly claims the line reaches Q at t=1. The setup and step 2 are correct, but this error invalidates the solution.qwen3.6:27b-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.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — 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.gpt-oss:20b: fail (error) 2026-10-04 — Step 3 incorrectly states the direction vector as (-2,-1,-1) instead of the correct (-2,-5,-1), and thus incorrectly claims the line reaches Q at t=1. The setup and step 2 are correct, but this error invalidates the solution.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the direction vector and the parametric equations. The explanation for deriving the symmetric form is mathematically sound.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.