Projections and orthogonality
Problem 9.264 · hard
Write \( \displaystyle \mathbf u = \left\langle -5, -1 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle -1, 5 \right\rangle \) and a vector orthogonal to it.
- \[ -5 + 5 = 0 \]u·v.✓ Proved
- \[ 1 + 25 = 26 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}0\\0\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
- \[ \left[\begin{matrix}-5\\-1\end{matrix}\right] \]The orthogonal part is u − proj.✓ Proved
- \[ 0 \]It is orthogonal to v.✓ Proved
Answer \( \left\langle 0, 0 \right\rangle + \left\langle -5, -1 \right\rangle \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 identifies that u is orthogonal to v, making the parallel component the zero vector and the orthogonal component u itself. The steps are logically sound and the conclusion is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies that u is orthogonal to v, making the parallel component the zero vector and the orthogonal component u itself. The steps are logically sound and the conclusion is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies that u is orthogonal to v, making the parallel component the zero vector and the orthogonal component u itself. The calculations and logic are sound.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.