Vectors in three dimensions
Problem 9.525 · medium
Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 12 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle -2, -6, -9 \right\rangle \).
- \[ 11 \]‖v‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{24}{11}\\- \frac{72}{11}\\- \frac{108}{11}\end{matrix}\right] \]12·v/‖v‖.✓ Proved
Answer \( \left\langle - \frac{24}{11}, - \frac{72}{11}, - \frac{108}{11} \right\rangle \)
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 checked, and parallel by the equality case of Cauchy–Schwarz |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly calculates the magnitude of v as 11 and scales the vector to have length 12. The algebraic steps are verified and the final answer is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the magnitude of v as 11 and scales the vector to have length 12. The algebraic steps are verified and the final answer is correct.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly states that the magnitude of v is 11. The actual magnitude is sqrt(4 + 36 + 81) = sqrt(121) = 11, so the calculation is numerically correct, but the first line is labeled 'unchecked' and presented as a fact without derivation, which is acceptable if the CAS checked it, but the prompt says '11 = 11 -- ||v||'. Wait, sqrt(121) is 11. So the math is correct. Let me re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. The lines provided are equations. The prompt says 'do not re-check the algebra of a line marked proved'. But these are marked 'unchecked'. However, the user instruction says '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 'unchecked'. This implies I should check them? Or does 'unchecked' mean the CAS didn't check it? The prompt says 'Each line... is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. It seems the input format is slightly ambiguous. If the line is an equation, it is assumed checked. If it is unchecked, maybe it's a sentence? No, it's clearly an equation. Let's assume the equations are correct. The math is: ||v|| = 11. u = 12/11 * v. Components: -24/11, -72/11, -108/11. This is correct. Why would it fail? Ah, look at the first line: '11 = 11 -- ||v||'. This is a tautology. It doesn't show the calculation of ||v||. But the second line shows the final vector. Is there a missing step? The problem asks to find the vector. The solution provides the vector. The intermediate step '11=11' is weird but not wrong. Is there a subtle error? No. The answer is correct. The solution is sound. Wait, is the prompt asking me to find what is wrong? 'Your job is to find what is wrong'. If nothing is wrong, say pass. Is '11=11' misleading? It asserts the norm is 11. It is correct. I will 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/vectors_space, checked 2026-10-11 with SymPy 1.14.0.