Home›Calculus 3›Projections and orthogonality›Problem 9.504

Projections and orthogonality

Problem 9.504 · easy

Which pairs of \( \displaystyle \mathbf a = \left\langle 4, 1, 3 \right\rangle \), \( \displaystyle \mathbf b = \left\langle -15, -3, 21 \right\rangle \), \( \displaystyle \mathbf c = \left\langle 3, -4, -5 \right\rangle \) are orthogonal?

\[ \left[\begin{matrix}0\\-7\\-138\end{matrix}\right] \]
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 computes the dot products and applies the definition of orthogonality. The conclusion that only a and b are orthogonal follows directly from the computed values.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly computes the dot products and applies the definition of orthogonality. The conclusion that only a and b are orthogonal follows directly from the computed values.
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 but does not conclude that only a and b are orthogonal because their dot product is 0, while the others are not. The stated answer is correct, but the solution text is incomplete and does not logically bridge the calculation to the final 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.