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Triple products and volumes

Problem 9.484 · medium

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

\[ \left[\begin{matrix}-9\\0\\18\end{matrix}\right] \]
u × v is orthogonal to both.✓ Proved
\[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{5}}{5}\\0\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{5}}{5}\\0\\\frac{2 \sqrt{5}}{5}\end{matrix}\right] \]
Divide by its length.✓ Proved

Answer \( \left\langle - \frac{\sqrt{5}}{5}, 0, \frac{2 \sqrt{5}}{5} \right\rangle\ (\text{or its negative}) \)

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 normalizes it to obtain a unit vector. The steps are logically sound and the final answer matches the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the cross product to find an orthogonal vector and normalizes it to obtain a unit vector. The steps are logically sound and the final answer matches the stated answer.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The cross product calculation is incorrect; the actual cross product of u and v is <9, 2, 18>, not <-9, 0, 18>. The y-component is non-zero, so the stated answer is not orthogonal to u.
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.