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Vectors in three dimensions

Problem 9.286 · easy

For \( \displaystyle \mathbf a = \left\langle 6, 3, 1 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle -3, 3, -6 \right\rangle \), find \( \displaystyle -4\mathbf a - (-2)\mathbf b \) and \( \displaystyle \|\mathbf a - \mathbf b\| \).

\[ \left[\begin{matrix}-30\\-6\\-16\end{matrix}\right] \]
Componentwise.✓ Proved
\[ \sqrt{130} \]
‖a − b‖.✓ Proved

Answer \( \left\langle -30, -6, -16 \right\rangle,\ \sqrt{130} \)

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	floating point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution provides no derivation or calculation steps, only tautological checks of the final answers. It fails to demonstrate how the results were obtained, which is the core requirement of a calculus solution.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution provides no derivation or calculation steps, only tautological checks of the final answers. It fails to demonstrate how the results were obtained, which is the core requirement of a calculus solution.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution provides no derivation or calculation steps, only tautological checks of the final answers. It fails to demonstrate how the vector operations and norm were computed.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.