Triple products and volumes
Problem 9.322 · medium
Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle 4, -2, -1 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle 4, 3, 4 \right\rangle \).
- \[ \left[\begin{matrix}-5\\-20\\20\end{matrix}\right] \]u × v is orthogonal to both.✓ Proved
- \[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{33}}{33}\\- \frac{4 \sqrt{33}}{33}\\\frac{4 \sqrt{33}}{33}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{33}}{33}\\- \frac{4 \sqrt{33}}{33}\\\frac{4 \sqrt{33}}{33}\end{matrix}\right] \]Divide by its length.✓ Proved
Answer \( \left\langle - \frac{\sqrt{33}}{33}, - \frac{4 \sqrt{33}}{33}, \frac{4 \sqrt{33}}{33} \right\rangle\ (\text{or its negative}) \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the cross product as the method to find an orthogonal vector and normalizes it to obtain a unit vector. The comments accurately describe the steps.
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 the method to find an orthogonal vector and normalizes it to obtain a unit vector. The comments accurately describe the steps.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The cross product calculation is incorrect. The actual cross product of <4, -2, -1> and <4, 3, 4> is <-5, -20, 20>, but the solution claims the result is <-5, -20, 20> without showing the work, and more importantly, the normalization in step 2 is based on the vector <-5, -20, 20>. The magnitude of <-5, -20, 20> is sqrt(25 + 400 + 400) = sqrt(825) = 5*sqrt(33). Dividing <-5, -20, 20> by 5*sqrt(33) gives <-1/sqrt(33), -4/sqrt(33), 4/sqrt(33)>, which simplifies to <-sqrt(33)/33, -4*sqrt(33)/33, 4*sqrt(33)/33>. Wait, let me re-check the cross product. u = <4, -2, -1>, v = <4, 3, 4>. i component: (-2)(4) - (-1)(3) = -8 + 3 = -5. j component: -((4)(4) - (-1)(4)) = -(16 + 4) = -20. k component: (4)(3) - (-2)(4) = 12 + 8 = 20. So the cross product is indeed <-5, -20, 20>. The magnitude is sqrt(25 + 400 + 400) = sqrt(825) = sqrt(25 * 33) = 5*sqrt(33). The unit vector is <-5/(5*sqrt(33)), -20/(5*sqrt(33)), 20/(5*sqrt(33))> = <-1/sqrt(33), -4/sqrt(33), 4/sqrt(33)>. Rationalizing the denominator: <-sqrt(33)/33, -4*sqrt(33)/33, 4*sqrt(33)/33>. The stated answer matches this. However, the solution text says 'u x v is orthogonal to both' and then just presents the matrix equality. It doesn't explicitly state the cross product formula or the calculation steps, but the equation is marked 'unchecked' by the prompt instructions? No, the prompt says 'Each line of the solution is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. The lines provided are labeled '[equation, unchecked]'. This is contradictory. If they are equations, they are checked. If they are unchecked, I should check them. Let's assume the label 'unchecked' means I need to verify the math. The math in the equations is correct. The sentences are '-- u x v is orthogonal to both.' and '-- Divide by its length.'. These are correct statements. Why did I think it was an error? Let me re-read carefully. Ah, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The lines provided are: '1. [equation, unchecked] ... -- u x v is orthogonal to both.' and '2. [equation, unchecked] ... -- Divide by its length.'. The text after the equation is the sentence. The sentence 'u x v is orthogonal to both' is correct. The sentence 'Divide by its length' is correct. The equations are correct. So why fail? Maybe the 'unchecked' tag implies I should check the equation? If I check the equation, it is correct. Is there a subtle error? The problem asks for 'a unit vector'. The solution provides one. The negative is also mentioned in the stated answer. The solution seems correct. Let me look for style issues. The solution is very sparse. It doesn't show the cross product calculation. But the equation line shows the result. Is the result correct? Yes. Is the normalization correct? Yes. Is the sentence misleading? No. Is it an error? No. Maybe I should pass it. But wait, the prompt says 'find what is wrong'. If there is nothing wrong, I pass. Let me double check the cross product again. u = <4, -2, -1>, v = <4, 3, 4>. i: (-2)(4) - (-1)(3) = -8 + 3 = -5. j: - (4*4 - (-1)*4) = - (16 + 4) = -20. k: 4*3 - (-2)*4 = 12 + 8 = 20. Vector: <-5, -20, 20>. Length: sqrt(25 + 400 + 400) = sqrt(825) = 5*sqrt(33). Unit vector: <-5/(5sqrt33), -20/(5sqrt33), 20/(5sqrt33)> = <-1/sqrt33, -4/sqrt33, 4/sqrt33>. Rationalized: <-sqrt33/33, -4sqrt33/33, 4sqrt33/33>. This matches the stated answer. The solution is correct. I will pass it.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.