Triple products and volumes
Problem 9.324 · easy
Are \( \displaystyle \mathbf u = \left\langle 2, -4, 2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -4, -4, -2 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle 4, -4, 1 \right\rangle \) coplanar?
- \[ \left[\begin{matrix}-12\\-4\\32\end{matrix}\right] \]v × w.✓ Proved
- \[ 56 \]u·(v × w).✓ Proved
- Three vectors are coplanar exactly when u·(v × w) = 0.Reviewed
Answer \( \text{not coplanar} \)
✓ 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the 3 × 3 determinant, computed by mpmath |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the scalar triple product test for coplanarity. Since the result is non-zero, the vectors are not coplanar, which matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the scalar triple product test for coplanarity. Since the result is non-zero, the vectors are not coplanar, which matches the stated answer.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the scalar triple product as the test for coplanarity and correctly concludes that a non-zero result implies the vectors are not coplanar.gpt-oss:20b: pass 2026-10-07
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/triple_product, checked 2026-10-07 with SymPy 1.14.0.