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Triple products and volumes

Problem 9.327 · medium

Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle 2, -1, 1 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle -2, -4, 3 \right\rangle \).

\[ \left[\begin{matrix}1\\-8\\-10\end{matrix}\right] \]
u × v is orthogonal to both.✓ Proved
\[ \left[\begin{matrix}\frac{\sqrt{165}}{165}\\- \frac{8 \sqrt{165}}{165}\\- \frac{2 \sqrt{165}}{33}\end{matrix}\right] \]
Divide by its length.✓ Proved

Answer \( \left\langle \frac{\sqrt{165}}{165}, - \frac{8 \sqrt{165}}{165}, - \frac{2 \sqrt{165}}{33} \right\rangle\ (\text{or its negative}) \)

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 to find an orthogonal vector and then normalizes it. The steps are logically sound and the final answer matches the stated answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly computes the cross product to find an orthogonal vector and then normalizes it. The steps are logically sound and the final answer matches the stated answer.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The cross product calculation is incorrect. The actual cross product of <2, -1, 1> and <-2, -4, 3> is <-1, -8, -6>, not <1, -8, -10>. Consequently, the final unit vector is wrong.
gpt-oss:20b: pass 2026-10-07

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 2026-10-07 with SymPy 1.14.0.