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Triple products and volumes

Problem 9.243 · easy

Are \( \displaystyle \mathbf u = \left\langle -1, 2, -1 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 2, 3, 4 \right\rangle \) and \( \displaystyle \mathbf w = \left\langle -2, -10, -6 \right\rangle \) coplanar?

\[ \left[\begin{matrix}22\\4\\-14\end{matrix}\right] \]
v × w.✓ Proved
\[ 0 \]
u·(v × w).✓ Proved
Three vectors are coplanar exactly when u·(v × w) = 0.
Reviewed

Answer \( \text{coplanar} \)

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 computes the scalar triple product and applies the standard theorem for coplanarity. The logic is sound and the conclusion follows directly.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the scalar triple product and applies the standard theorem for coplanarity. The logic is sound and the conclusion follows directly.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The cross product v × w is calculated incorrectly; the correct value is <56, -20, -4>, not <22, 4, -14>. Although the final conclusion is correct because u is indeed a linear combination of v and w, the intermediate calculation is wrong, which would teach a student incorrect arithmetic.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.