Projections and orthogonality
Problem 9.422 · medium
Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -4, 2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -3, -5 \right\rangle \).
- \[ -10 + 12 = 2 \]u·v.✓ Proved
- \[ 9 + 25 = 34 \]‖v‖².✓ Proved
- \[ \frac{\sqrt{34}}{17} \]comp = u·v / ‖v‖.✓ Proved
Answer \( \frac{\sqrt{34}}{17} \)
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 | ‖u‖ cos θ with θ from arccos |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The scalar projection is defined as (u·v) / ||v||. The solution correctly calculates u·v = 2 and ||v||² = 34, but then incorrectly states the final r
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The scalar projection is defined as (u·v) / ||v||. The solution correctly calculates u·v = 2 and ||v||² = 34, but then incorrectly states the final rgpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution calculates the scalar projection as 1/sqrt(34) instead of the correct value 2/sqrt(34). This is because it uses a numerator of 1 (implied by the result) instead of the actual dot product u·v = 2.
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.