Lines in space
Problem 9.469 · easy
Find parametric and symmetric equations of the line through \( \displaystyle P(0, 5, -5) \) and \( \displaystyle Q(-5, 1, 6) \).
- \[ \left[\begin{matrix}-5\\-4\\11\end{matrix}\right] \]A direction vector.✓ Proved
- r(t) = P + t·(Q − P); solving each coordinate for t gives the symmetric form.Reviewed
- \[ \left[\begin{matrix}-5\\1\\6\end{matrix}\right] \]At t = 1 the line reaches Q.✓ Proved
Answer \( x = - 5 t,\ y = 5 - 4 t,\ z = 11 t - 5;\quad \frac{x - (0)}{-5} = \frac{y - (5)}{-4} = \frac{z - (-5)}{11} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 formula for parametric and symmetric equations of a line. The logic is sound and the final answer matches the derived equations.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the direction vector and applies the standard formula for parametric and symmetric equations of a line. The logic is sound and the final answer matches the derived equations.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly identifies the direction vector and the parametric equations. The explanation for deriving the symmetric form is mathematically sound, and the final answer matches the derived equations.
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 2026-10-10 with SymPy 1.14.0.