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Triple products and volumes

Problem 9.486 · easy

Are \( \displaystyle \mathbf u = \left\langle 1, -1, 2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -2, -1, -4 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle 4, -1, 0 \right\rangle \) coplanar?

\[ \left[\begin{matrix}-4\\-16\\6\end{matrix}\right] \]
v × w.✓ Proved
\[ 24 \]
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: pass
qwen3.6:27b-mlx: fail (error) — The solution correctly calculates the scalar triple product as 24, but fails to explicitly state that since 24 is not equal to 0, the vectors are not coplanar. It stops at the definition without applying it to the computed value.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution correctly calculates the scalar triple product as 24, but fails to explicitly state that since 24 is not equal to 0, the vectors are not coplanar. It stops at the definition without applying it to the computed value.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution calculates the scalar triple product as 24, which is non-zero, implying the vectors are not coplanar. However, it fails to explicitly state this conclusion or connect the non-zero result to the final answer, leaving the logical gap between the calculation and the stated answer.
gpt-oss:20b: pass 2026-10-11

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 2026-10-11 with SymPy 1.14.0.