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Triple products and volumes

Problem 9.482 · medium

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

\[ \left[\begin{matrix}-12\\6\\0\end{matrix}\right] \]
u × v is orthogonal to both.✓ Proved
\[ \left[\begin{matrix}- \frac{2 \sqrt{5}}{5}\\\frac{\sqrt{5}}{5}\\0\end{matrix}\right] \]
Divide by its length.✓ Proved

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

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 then 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 then 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 <2, 4, 1> and <-2, -4, -4> is <-12, 6, 0>, but the solution implies this vector is already normalized or skips the normalization step entirely in the text, while the second line shows a normalized vector that does not correspond to the unnormalized vector in line 1 without an explicit division step shown in the text. More critically, the vector in line 2 is not a unit vector; its magnitude is sqrt(4/5 + 1/5) = 1, wait, let me recheck. (-2sqrt(5)/5)^2 + (sqrt(5)/5)^2 = 4*5/25 + 5/25 = 20/25 + 5/25 = 1. So it is a unit vector. However, the transition from line 1 to line 2 is missing the normalization calculation. The vector in line 1 is <-12, 6, 0>. Its length is sqrt(144 + 36) = sqrt(180) = 6*sqrt(5). Dividing <-12, 6, 0> by 6*sqrt(5) gives <-2/sqrt(5), 1/sqrt(5), 0> which is <-2sqrt(5)/5, sqrt(5)/5, 0>. So the math is correct. Why did I think it was wrong? Let me re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is an equation. Line 2 is an equation. The text '-- u x v is orthogonal to both' is attached to line 1. The text '-- Divide by its length' is attached to line 2. The solution is actually correct. Let me double check the cross product. u = <2, 4, 1>, v = <-2, -4, -4>. i(4*-4 - 1*-4) - j(2*-4 - 1*-2) + k(2*-4 - 4*-2). i(-16 + 4) - j(-8 + 2) + k(-8 + 8). i(-12) - j(-6) + k(0) = <-12, 6, 0>. Correct. Length sqrt(144+36)=sqrt(180)=6sqrt(5). Unit vector <-12/6sqrt(5), 6/6sqrt(5), 0> = <-2/sqrt(5), 1/sqrt(5), 0>. Rationalized: <-2sqrt(5)/5, sqrt(5)/5, 0>. The solution is correct. Wait, is there a subtle error? The problem asks for 'a unit vector'. The answer provides one. The steps are logically sound. Is it 'style' because it doesn't show the intermediate normalization step? The prompt says 'find what is wrong'. If it's correct, I should say pass. But usually these adversarial reviews have a trap. Let me look closer at the vectors. u = <2, 4, 1>. v = <-2, -4, -4>. Are they parallel? No. Is the cross product zero? No. Is the result orthogonal? <-12, 6, 0> dot <2, 4, 1> = -24 + 24 + 0 = 0. <-12, 6, 0> dot <-2, -4, -4> = 24 - 24 + 0 = 0. Yes. Is the final vector a unit vector? Yes. Is it orthogonal? Yes. Why would this fail? Maybe the 'unchecked' tag on the equations means I shouldn't check the algebra? 'do not re-check the algebra of a line marked proved'. Here it says 'unchecked'. This usually means the CAS hasn't verified it, or I should verify it? The prompt says 'Each line... is either an EQUATION, which a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. The lines are marked '[equation, unchecked]'. This is contradictory. If it's unchecked, I should check it? Or does 'unchecked' mean the CAS didn't check it? The prompt says 'do not re-check the algebra of a line marked proved'. These are marked 'unchecked'. So I should check the algebra? If I check the algebra, it is correct. So why fail? Maybe the sentence '-- Divide by its length' is misleading because it doesn't specify *which* length? No, it's clear. Maybe the issue is that the solution doesn't explicitly state that the cross product is the method used? Line 1 says 'u x v is orthogonal to both'. That's the method. I see no error. I will mark it as pass.
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.