Triple products and volumes
Problem 9.323 · medium
Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle 4, 1, 2 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle 1, 2, -3 \right\rangle \).
- \[ \left[\begin{matrix}-7\\14\\7\end{matrix}\right] \]u × v is orthogonal to both.✓ Proved
- \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\\\frac{\sqrt{6}}{6}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{6}}{6}\\\frac{\sqrt{6}}{3}\\\frac{\sqrt{6}}{6}\end{matrix}\right] \]Divide by its length.✓ Proved
Answer \( \left\langle - \frac{\sqrt{6}}{6}, \frac{\sqrt{6}}{3}, \frac{\sqrt{6}}{6} \right\rangle\ (\text{or its negative}) \)
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 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)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — 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.qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The cross product calculation is incorrect; the correct cross product of <4, 1, 2> and <1, 2, -3> is <-7, 14, 7>, but the solution presents this asgpt-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.