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Vectors in the plane

Problem 9.358 · easy

For \( \displaystyle \mathbf a = \left\langle 6, 3 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle 5, -3 \right\rangle \), find \( \displaystyle -4\mathbf a - 3\mathbf b \) and its magnitude.

\[ \left[\begin{matrix}-39\\-3\end{matrix}\right] \]
Scale each vector, then add componentwise.✓ Proved
\[ 3 \sqrt{170} \]
Its length.✓ Proved

Answer \( \left\langle -39, -3 \right\rangle,\ \text{magnitude}\ 3 \sqrt{170} \)

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: pass — The solution correctly computes the linear combination and its magnitude. The comments accurately describe the steps taken.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly computes the linear combination and its magnitude. The comments accurately describe the steps taken.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution provides no intermediate steps showing the scalar multiplication and vector addition, making it impossible to verify the arithmetic or the method. Additionally, the 'equations' are tautologies (A=A) rather than derivations, which fails to demonstrate the calculation process.

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_plane, checked 2026-10-07 with SymPy 1.14.0.