Triple products and volumes
Problem 9.561 · easy
Are \( \displaystyle \mathbf u = \left\langle 0, 2, -2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -3, -2, -3 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle 0, 3, 0 \right\rangle \) coplanar?
- \[ \left[\begin{matrix}9\\0\\-9\end{matrix}\right] \]v × w.✓ Proved
- \[ 18 \]u·(v × w).✓ Proved
- Three vectors are coplanar exactly when u·(v × w) = 0.
Answer \( \text{not coplanar} \)
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | 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: fail (error) — The solution correctly calculates the scalar triple product as 18, but fails to conclude that the vectors are not coplanar because the product is non-zero. It states the condition for coplanarity but does not apply it to the calculated result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution correctly calculates the scalar triple product as 18, but fails to conclude that the vectors are not coplanar because the product is non-zero. It states the condition for coplanarity but does not apply it to the calculated result.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution calculates a non-zero scalar triple product (18) but fails to conclude that the vectors are not coplanar. It states the condition for coplanarity but does not apply it to the calculated result, leaving the logical connection to the final answer missing.
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 2026-10-13 with SymPy 1.14.0.