Projections and orthogonality
Problem 9.348 · hard
Write \( \displaystyle \mathbf u = \left\langle -6, -4, -5 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle 4, 4, -6 \right\rangle \) and a vector orthogonal to it.
- \[ -24 - 16 + 30 = -10 \]u·v.✓ Proved
- \[ 16 + 16 + 36 = 68 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}- \frac{10}{17}\\- \frac{10}{17}\\\frac{15}{17}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
- \[ \left[\begin{matrix}- \frac{92}{17}\\- \frac{58}{17}\\- \frac{100}{17}\end{matrix}\right] \]The orthogonal part is u − proj.✓ Proved
- \[ 0 \]It is orthogonal to v.✓ Proved
Answer \( \left\langle - \frac{10}{17}, - \frac{10}{17}, \frac{15}{17} \right\rangle + \left\langle - \frac{92}{17}, - \frac{58}{17}, - \frac{100}{17} \right\rangle \)
✓ 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 applies the vector projection formula to decompose u into parallel and orthogonal components. All algebraic steps are verified, and the orthogonality check confirms the result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the vector projection formula to decompose u into parallel and orthogonal components. All algebraic steps are verified, and the orthogonality check confirms the result.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly computes the projection and orthogonal components, and verifies orthogonality. The algebra is correct and the steps are logically sound.
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 2026-10-07 with SymPy 1.14.0.