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Triple products and volumes

Problem 9.483 · easy

Are \( \displaystyle \mathbf u = \left\langle 2, -3, -1 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -1, 0, -4 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle -1, -1, 4 \right\rangle \) coplanar?

\[ \left[\begin{matrix}-4\\8\\1\end{matrix}\right] \]
v × w.✓ Proved
\[ -33 \]
u·(v × w).✓ Proved
Three vectors are coplanar exactly when u·(v × w) = 0.
Reviewed

Answer \( \text{not coplanar} \)

Lines: 2 proved, 1 reviewed. 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	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: pass
qwen3.6:27b-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)

qwen3.6:27b-mlx: pass 2026-10-11 — 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.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution computes a non-zero scalar triple product (-33) but fails to explicitly conclude that the vectors are not coplanar. The final sentence only states the condition for coplanarity, omitting the logical deduction that since the product is not zero, the vectors are not coplanar.
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.