Lines in space
Problem 9.223 · easy
Find parametric and symmetric equations of the line through \( \displaystyle P(5, -3, -3) \) and \( \displaystyle Q(-4, -2, 0) \).
- \[ \left[\begin{matrix}-9\\1\\3\end{matrix}\right] \]A direction vector.✓ Proved
- r(t) = P + t·(Q − P); solving each coordinate for t gives the symmetric form.Reviewed
- \[ \left[\begin{matrix}-4\\-2\\0\end{matrix}\right] \]At t = 1 the line reaches Q.✓ Proved
Answer \( x = 5 - 9 t,\ y = t - 3,\ z = 3 t - 3;\quad \frac{x - (5)}{-9} = \frac{y - (-3)}{1} = \frac{z - (-3)}{3} \)
✓ 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 the parametric form. The sentence accurately describes the standard method for deriving symmetric equations from parametric ones, and the final answer matches the derived equations.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the direction vector and the parametric form. The sentence accurately describes the standard method for deriving symmetric equations from parametric ones, and the final answer matches the derived equations.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-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: 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.