Home›Calculus 3›Projections and orthogonality›Problem 9.510

Projections and orthogonality

Problem 9.510 · easy

Which pairs of \( \displaystyle \mathbf a = \left\langle -2, 1, 5 \right\rangle \), \( \displaystyle \mathbf b = \left\langle 24, -12, 12 \right\rangle \), \( \displaystyle \mathbf c = \left\langle 3, -5, 2 \right\rangle \) are orthogonal?

\[ \left[\begin{matrix}0\\-1\\156\end{matrix}\right] \]
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 applies it to the computed dot products. Although the final conclusion is not explicitly stated as a sentence, the provided lines are mathematically sound and sufficient to derive the answer.

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 applies it to the computed dot products. Although the final conclusion is not explicitly stated as a sentence, the provided lines are mathematically sound and sufficient to derive the answer.
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution calculates the dot products but fails to explicitly state which pairs are orthogonal based on the results. It lists the dot products (0, -1, 156) but does not conclude that only a and b are orthogonal because their dot product is 0, while the others are not.
gpt-oss:20b: pass 2026-10-11

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.