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Vectors in the plane

Problem 9.277 · easy

For \( \displaystyle \mathbf a = \left\langle -3, 6 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle 5, 5 \right\rangle \), find \( \displaystyle -4\mathbf a + 1\mathbf b \) and its magnitude.

\[ \left[\begin{matrix}17\\-19\end{matrix}\right] \]
Scale each vector, then add componentwise.✓ Proved
\[ 5 \sqrt{26} \]
Its length.✓ Proved

Answer \( \left\langle 17, -19 \right\rangle,\ \text{magnitude}\ 5 \sqrt{26} \)

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 vector combination and its magnitude. The comments accurately describe the operations performed.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the vector combination and its magnitude. The comments accurately describe the operations performed.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to show the intermediate calculation steps required to verify the vector arithmetic and magnitude. Specifically, it does not demonstrate that -4a + b equals <17, -19> or that the magnitude calculation is correct, making it impossible to verify the work.
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_plane, checked 2026-10-05 with SymPy 1.14.0.