Projections and orthogonality
Problem 9.501 · medium
Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -2, 6, -2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -6, -5, -6 \right\rangle \).
- \[ -30 + 12 + 12 = -6 \]u·v.✓ Proved
- \[ 25 + 36 + 36 = 97 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}\frac{36}{97}\\\frac{30}{97}\\\frac{36}{97}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
Answer \( \left\langle \frac{36}{97}, \frac{30}{97}, \frac{36}{97} \right\rangle \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the remainder is orthogonal to v and the projection is parallel to it |
Reviewers
gpt-oss:20b: fail (error) — The dot product u·v is -30, so the scalar factor is -30/97. Multiplying this by v = ⟨-6,-5,-6⟩ gives ⟨180/97,150/97,180/97⟩, not ⟨36/97,30/97,36/97⟩. The solution’s projection vector is incorrect.qwen3.6:27b-mlx: pass — The solution correctly applies the vector projection formula. The dot product and norm squared calculations are correct, and the final vector matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-11 — The dot product u·v is -30, so the scalar factor is -30/97. Multiplying this by v = ⟨-6,-5,-6⟩ gives ⟨180/97,150/97,180/97⟩, not ⟨36/97,30/97,36/97⟩. The solution’s projection vector is incorrect.qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the vector projection formula. The dot product and norm squared calculations are correct, and the final vector matches the stated answer.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the vector projection formula. The dot product and squared norm are calculated correctly, and the final vector is derived accurately.
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.