Triple products and volumes
Problem 9.321 · medium
Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle -3, 0, 4 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle -2, 1, 0 \right\rangle \).
- \[ \left[\begin{matrix}-4\\-8\\-3\end{matrix}\right] \]u × v is orthogonal to both.✓ Proved
- \[ \left[\begin{matrix}- \frac{4 \sqrt{89}}{89}\\- \frac{8 \sqrt{89}}{89}\\- \frac{3 \sqrt{89}}{89}\end{matrix}\right] \]Divide by its length.✓ Proved
Answer \( \left\langle - \frac{4 \sqrt{89}}{89}, - \frac{8 \sqrt{89}}{89}, - \frac{3 \sqrt{89}}{89} \right\rangle\ (\text{or its negative}) \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the cross product as orthogonal to both vectors and normalizes it to obtain a unit vector. The algebraic steps are verified, and the final result matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the cross product as orthogonal to both vectors and normalizes it to obtain a unit vector. The algebraic steps are verified, and the final result matches the stated answer.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The cross product calculation is incorrect. The actual cross product of u and v is < -4, -8, 3 >, but the solution uses < -4, -8, -3 >, resulting in a vector with the wrong z-component sign.
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.