Projections and orthogonality
Problem 9.426 · easy
Which pairs of \( \displaystyle \mathbf a = \left\langle 4, 2, -1 \right\rangle \), \( \displaystyle \mathbf b = \left\langle -1, -2, -8 \right\rangle \), \( \displaystyle \mathbf c = \left\langle -4, -2, -1 \right\rangle \) are orthogonal?
- \[ \left[\begin{matrix}0\\-19\\16\end{matrix}\right] \]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: passqwen3.6:27b-mlx: pass — The solution correctly computes the dot products and applies the definition of orthogonality to identify the correct pair.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the dot products and applies the definition of orthogonality to identify the correct pair.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution computes the dot products correctly but fails to identify that a·c = 0, meaning vectors a and c are also orthogonal. The stated answer is incomplete.
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 2026-10-08 with SymPy 1.14.0.