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Projections and orthogonality

Problem 9.509 · easy

Which pairs of \( \displaystyle \mathbf a = \left\langle -3, -1, 1 \right\rangle \), \( \displaystyle \mathbf b = \left\langle -2, -3, -9 \right\rangle \), \( \displaystyle \mathbf c = \left\langle -1, 4, -1 \right\rangle \) are orthogonal?

\[ \left[\begin{matrix}0\\-2\\-1\end{matrix}\right] \]
a·b, a·c, b·c.✓ Proved
Two nonzero vectors are orthogonal exactly when their dot product is 0.
Reviewed

Answer \( \mathbf a \perp \mathbf b \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	dot products recomputed

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the orthogonality condition and the computed dot products confirm that only a and b are orthogonal.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the orthogonality condition and the computed dot products confirm that only a and b are orthogonal.
qwen3.6:27b-mlx: inconclusive 2026-10-11 — reviewer returned a non-object
gpt-oss:20b: fail (error) 2026-10-11 — The solution never shows that the dot product of α and β is actually zero; it merely lists the dot products in a matrix and repeats the same matrix. Thus the conclusion that α⊥β is unsupported.

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/vector_projection, checked 2026-10-11 with SymPy 1.14.0.