Projections and orthogonality
Problem 9.267 · easy
Which pairs of \( \displaystyle \mathbf a = \left\langle -4, 0, 6 \right\rangle \), \( \displaystyle \mathbf b = \left\langle -18, 30, -12 \right\rangle \), \( \displaystyle \mathbf c = \left\langle -6, 1, 3 \right\rangle \) are orthogonal?
- \[ \left[\begin{matrix}0\\42\\102\end{matrix}\right] \]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: passqwen3.6:27b-mlx: fail (error) — The solution computes the dot products correctly but fails to identify that a·b = 0, implying a is orthogonal to b. It incorrectly concludes a is orthogonal to b based on the first entry being 0, but the stated answer claims a ⊥ b while the matrix shows a·b=0, a·c=42, b·c=102. Wait, the stated answer is a ⊥ b. The matrix shows a·b = 0. So a and b are orthogonal. The solution text is missing the step that explicitly reads off the zero dot product to conclude orthogonality. However, looking closer at the prompt's 'Stated answer': a ⊥ b. The matrix shows a·b=0. So the answer is correct. But the solution text only states the definition. It does not explicitly state 'Since a·b=0, a and b are orthogonal'. Is this an error or just incomplete? The prompt asks to review the solution. The solution consists of an equation and a sentence. The sentence is a general theorem. It does not apply the theorem to the specific case. A student would not know which pair is orthogonal just from reading these two lines. The connection is missing. This is misleading/incomplete reasoning.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution computes the dot products correctly but fails to identify that a·b = 0, implying a is orthogonal to b. It incorrectly concludes a is orthogonal to b based on the first entry being 0, but the stated answer claims a ⊥ b while the matrix shows a·b=0, a·c=42, b·c=102. Wait, the stated answer is a ⊥ b. The matrix shows a·b = 0. So a and b are orthogonal. The solution text is missing the step that explicitly reads off the zero dot product to conclude orthogonality. However, looking closer at the prompt's 'Stated answer': a ⊥ b. The matrix shows a·b=0. So the answer is correct. But the solution text only states the definition. It does not explicitly state 'Since a·b=0, a and b are orthogonal'. Is this an error or just incomplete? The prompt asks to review the solution. The solution consists of an equation and a sentence. The sentence is a general theorem. It does not apply the theorem to the specific case. A student would not know which pair is orthogonal just from reading these two lines. The connection is missing. This is misleading/incomplete reasoning.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution calculates the dot products correctly (a·b=0, a·c=42, b·c=102) but fails to explicitly state which pairs are orthogonal based on these results. It provides the definition of orthogonality but does not apply it to conclude that only a and b are orthogonal, leaving the answer implicit and the logic incomplete.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/vector_projection, checked 2026-10-05 with SymPy 1.14.0.