Projections and orthogonality
Problem 9.427 · hard
Write \( \displaystyle \mathbf u = \left\langle 3, -2, -5 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle 3, -2, -6 \right\rangle \) and a vector orthogonal to it.
- \[ 4 + 9 + 30 = 43 \]u·v.✓ Proved
- \[ 4 + 9 + 36 = 49 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}\frac{129}{49}\\- \frac{86}{49}\\- \frac{258}{49}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
- \[ \left[\begin{matrix}\frac{18}{49}\\- \frac{12}{49}\\\frac{13}{49}\end{matrix}\right] \]The orthogonal part is u − proj.✓ Proved
- \[ 0 \]It is orthogonal to v.✓ Proved
Answer \( \left\langle \frac{129}{49}, - \frac{86}{49}, - \frac{258}{49} \right\rangle + \left\langle \frac{18}{49}, - \frac{12}{49}, \frac{13}{49} \right\rangle \)
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly computes the projection and orthogonal components, and verifies orthogonality. The steps are logically sound and algebraically correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the projection and orthogonal components, and verifies orthogonality. The steps are logically sound and algebraically correct.qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The dot product calculation in step 1 is incorrect; u·v should be 9 + 4 + 30 = 43, but the calculation shown (4 + 9 + 30) implies incorrect componentgpt-oss:20b: pass 2026-10-08
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.