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Triple products and volumes

Problem 9.406 · medium

Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle -3, 1, 2 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle 2, 2, -3 \right\rangle \).

\[ \left[\begin{matrix}-7\\-5\\-8\end{matrix}\right] \]
u × v is orthogonal to both.✓ Proved
\[ \left[\begin{matrix}- \frac{7 \sqrt{138}}{138}\\- \frac{5 \sqrt{138}}{138}\\- \frac{4 \sqrt{138}}{69}\end{matrix}\right] \]
Divide by its length.✓ Proved

Answer \( \left\langle - \frac{7 \sqrt{138}}{138}, - \frac{5 \sqrt{138}}{138}, - \frac{4 \sqrt{138}}{69} \right\rangle\ (\text{or its negative}) \)

Every line of this solution was proved by the computer algebra system SymPy. 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	length 1 and zero dot product with u and v

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the cross product and normalizes it to find a unit vector orthogonal to both given vectors. The steps are logically sound and the final answer matches the stated answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the cross product and normalizes it to find a unit vector orthogonal to both given vectors. The steps are logically sound and the final answer matches the stated answer.
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The cross product calculation is incorrect; the z-component should be -8, not -4, leading to an incorrect final vector.
gpt-oss:20b: pass 2026-10-08

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 2026-10-08 with SymPy 1.14.0.