Triple products and volumes
Problem 9.403 · medium
Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle -4, 3, -4 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle -2, 2, -3 \right\rangle \).
- \[ \left[\begin{matrix}-1\\-4\\-2\end{matrix}\right] \]u × v is orthogonal to both.✓ Proved
- \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{21}}{21}\\- \frac{4 \sqrt{21}}{21}\\- \frac{2 \sqrt{21}}{21}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{21}}{21}\\- \frac{4 \sqrt{21}}{21}\\- \frac{2 \sqrt{21}}{21}\end{matrix}\right] \]Divide by its length.✓ Proved
Answer \( \left\langle - \frac{\sqrt{21}}{21}, - \frac{4 \sqrt{21}}{21}, - \frac{2 \sqrt{21}}{21} \right\rangle\ (\text{or its negative}) \)
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. A reviewer's objection to the explanation stands; see 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: fail (error) — The solution never actually computes the cross product of u and v; it simply states a vector that is not shown to be u×v. The claim that this vector is orthogonal to both u and v is unsupported, so the setup is incorrect.qwen3.6:27b-mlx: fail (error) — The first equation claims the cross product is <-1, -4, -2>, but the actual cross product of u and v is <-1, -10, -4>. The setup is numerically incorrect.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — The solution never actually computes the cross product of u and v; it simply states a vector that is not shown to be u×v. The claim that this vector is orthogonal to both u and v is unsupported, so the setup is incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The first equation claims the cross product is <-1, -4, -2>, but the actual cross product of u and v is <-1, -10, -4>. The setup is numerically incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to compute the cross product explicitly, jumping directly to the final normalized vector. Without showing the intermediate cross product vector, the student cannot verify that the direction is correct or that the normalization was performed on the correct vector.gpt-oss:20b: fail (error) 2026-10-08 — The cross‑product was computed with a sign error: the second component should be +4, not –4. Consequently the unit vector’s second component is also wrong. The setup is correct, but the algebraic result is incorrect.
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.