Triple products and volumes
Problem 9.568 · medium
Find a unit vector orthogonal to both \( \displaystyle \mathbf u = \left\langle 0, 3, -3 \right\rangle \) and \( \displaystyle \mathbf v = \left\langle 2, 4, -2 \right\rangle \).
- \[ \left[\begin{matrix}6\\-6\\-6\end{matrix}\right] \]u × v is orthogonal to both.✓ Proved
- \[ \left[\begin{matrix}\frac{\sqrt{3}}{3}\\\frac{\left(-1\right) \sqrt{3}}{3}\\\frac{\left(-1\right) \sqrt{3}}{3}\end{matrix}\right] = \left[\begin{matrix}\frac{\sqrt{3}}{3}\\- \frac{\sqrt{3}}{3}\\- \frac{\sqrt{3}}{3}\end{matrix}\right] \]Divide by its length.✓ Proved
Answer \( \left\langle \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3} \right\rangle\ (\text{or its negative}) \)
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 final answer matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly computes the cross product to find an orthogonal vector and normalizes it to obtain a unit vector. The final answer matches the stated answer.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The cross product calculation is incorrect. The actual cross product of <0, 3, -3> and <2, 4, -2> is <6, -6, -6>, but the normalization step is flawed in the explanation (though the final vector happens to be correct, the intermediate vector shown in step 1 is not normalized, and step 2 claims to divide by length but shows the result directly without showing the magnitude calculation or confirming the vector in step 1 is the one being normalized. More critically, the vector <6, -6, -6> has magnitude sqrt(36+36+36) = sqrt(108) = 6*sqrt(3). Dividing <6, -6, -6> by 6*sqrt(3) gives <1/sqrt(3), -1/sqrt(3), -1/sqrt(3)> which simplifies to <sqrt(3)/3, -sqrt(3)/3, -sqrt(3)/3>. The solution skips the magnitude calculation and the division step, jumping from the unnormalized vector to the normalized one, which is a logical gap. However, the primary issue is that the solution presents the unnormalized vector in step 1 and the normalized vector in step 2 without showing the work, which is acceptable for a summary but the prompt asks to review sentences. The sentences are minimal. Let's look closer. Step 1 says 'u x v is orthogonal'. Step 2 says 'Divide by its length'. The equations are marked unchecked but the prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Wait, the lines are marked 'unchecked'. This implies I should check the algebra? No, the prompt says 'Each line... is either an EQUATION... or a SENTENCE... The sentences are what you are here for.' It also says 'do not re-check the algebra of a line marked proved'. These lines are marked 'unchecked'. This is ambiguous. Usually, 'unchecked' means the CAS hasn't verified it, or it's a statement. But the content is an equation. If I assume the equations are correct (as per 'do not re-check... marked proved' - wait, they are NOT marked proved, they are marked unchecked. This likely means I SHOULD check them or they are just placeholders. Let's look at the vectors. u x v = <6, -6, -6>. The vector in step 1 is <6, -6, -6>. This is correct. The vector in step 2 is <sqrt(3)/3, ...>. This is the unit vector. The sentence 'Divide by its length' is the justification. Is the length of <6, -6, -6> equal to the factor needed? Length is 6*sqrt(3). 6 / (6*sqrt(3)) = 1/sqrt(3) = sqrt(3)/3. The math is correct. The sentences are sparse but correct. Why would it fail? Maybe the direction? 'orthogonal to both'. Cross product gives one. Negative is also valid. The answer includes 'or its negative'. The solution provides one. This is standard. Is there a subtle error? u = <0, 3, -3>, v = <2, 4, -2>. i(3*-2 - (-3)*4) = i(-6 + 12) = 6i. j(0*-2 - (-3)*2) = j(0 + 6) = 6j, but cross product formula is -j(...), so -6j. k(0*4 - 3*2) = -6k. Result <6, -6, -6>. Correct. Normalization: sqrt(36+36+36) = sqrt(108) = 6sqrt(3). Unit vector: <6/6sqrt(3), -6/6sqrt(3), -6/6sqrt(3)> = <1/sqrt(3), ...> = <sqrt(3)/3, ...>. Correct. The solution is sound. The 'unchecked' tag might be a distractor or imply I should check the algebra if it's not 'proved'. But the instructions say 'do not re-check the algebra of a line marked proved'. It doesn't explicitly forbid checking unchecked lines, but it says 'The sentences are what you are here for'. The sentences are 'u x v is orthogonal to both' and 'Divide by its length'. These are correct statements. I will pass it.gpt-oss:20b: pass 2026-10-13
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 2026-10-13 with SymPy 1.14.0.