Home›Calculus 3›Projections and orthogonality›Problem 9.503

Projections and orthogonality

Problem 9.503 · easy

Which pairs of \( \displaystyle \mathbf a = \left\langle -4, -2, 1 \right\rangle \), \( \displaystyle \mathbf b = \left\langle 14, -13, 30 \right\rangle \), \( \displaystyle \mathbf c = \left\langle -3, -6, 5 \right\rangle \) are orthogonal?

\[ \left[\begin{matrix}0\\29\\186\end{matrix}\right] \]
a·b, a·c, b·c.✓ Proved
Two nonzero vectors are orthogonal exactly when their dot product is 0.

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

Lines: 1 proved, 1 not checked. 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	Not checked	—	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: fail (error) — The solution computes the dot products but fails to explicitly state which pairs are orthogonal based on the results. It lists the dot products as 0, 29, and 186, but does not conclude that only a and b are orthogonal, leaving the answer incomplete.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution computes the dot products but fails to explicitly state which pairs are orthogonal based on the results. It lists the dot products as 0, 29, and 186, but does not conclude that only a and b are orthogonal, leaving the answer incomplete.
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 support the stated answer.

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.