Home›Calculus 3›Vectors in three dimensions›Problem 9.609

Vectors in three dimensions

Problem 9.609 · medium

Find the vector \( \displaystyle \mathbf u \) with \( \displaystyle \|\mathbf u\| = 12 \) pointing in the same direction as \( \displaystyle \mathbf v = \left\langle 4, -4, 7 \right\rangle \).

\[ 9 \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}\frac{16}{3}\\- \frac{16}{3}\\\frac{28}{3}\end{matrix}\right] \]
12·v/‖v‖.✓ Proved

Answer \( \left\langle \frac{16}{3}, - \frac{16}{3}, \frac{28}{3} \right\rangle \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly calculates the magnitude of v as 9 and scales v by 12/9 to obtain the correct vector. The algebraic steps are verified.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-13
qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly calculates the magnitude of v as 9 and scales v by 12/9 to obtain the correct vector. The algebraic steps are verified.
qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution claims the magnitude of v is 9, but the actual magnitude is sqrt(4^2 + (-4)^2 + 7^2) = sqrt(81) = 9. Wait, 16+16+49 = 81, so sqrt(81) is indeed 9. The calculation is correct. Why did I think it was wrong? Let me re-read. Ah, the prompt says '9 = 9 -- ‖v‖'. This is a tautology, not a calculation. But more importantly, the final answer is correct. Is there a subtle error? No. The steps are: 1. Calculate norm. 2. Scale vector. The norm is 9. 12/9 = 4/3. 4 * 4/3 = 16/3. -4 * 4/3 = -16/3. 7 * 4/3 = 28/3. The math is correct. The 'sentence' part is just the comment '-- ‖v‖'. Is that misleading? No. It's just a label. Wait, looking at the JSON structure required. The user asks to review 'worked calculus solutions'. The solution provided is extremely sparse. It just shows two equations. Line 1 is '9=9'. This is not a derivation. It's a statement of fact. Line 2 is the final answer. Is it 'error'? No. Is it 'misleading'? It skips the step of showing how 9 was derived from v. A student might not know where 9 came from. However, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is an equation. It is marked 'unchecked'. The CAS checked it? 9=9 is true. The comment says '-- ‖v‖'. This implies the solver calculated ‖v‖ and got 9. This is correct. Line 2 is the final vector. The comment says '-- 12·v/‖v‖'. This implies the formula used. This is correct. So why would it fail? Maybe the 'unchecked' tag is suspicious? No, the instructions say 'do not re-check the algebra of a line marked proved'. Here it says 'unchecked'. Wait, 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 input says '[equation, unchecked]'. This contradicts the premise that equations are checked. Or does it mean the CAS *didn't* check it? If the CAS didn't check it, I should check it. 9=9 is trivially true. 12*v/9 = <16/3, -16/3, 28/3>. This is also true. So the math is correct. Is there a style issue? The solution is very brief. But is it 'wrong'? No. Let me look closer at the problem statement. 'Find the vector u...'. The answer is correct. The steps are logically sound, albeit brief. I will mark it as pass.
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/vectors_space, checked 2026-10-13 with SymPy 1.14.0.