Projections and orthogonality
Problem 9.507 · medium
Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -4, 5, -6 \right\rangle \), \( \displaystyle \mathbf v = \left\langle 6, 1, 2 \right\rangle \).
- \[ -24 - 12 + 5 = -31 \]u·v.✓ Proved
- \[ 1 + 4 + 36 = 41 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}- \frac{186}{41}\\- \frac{31}{41}\\- \frac{62}{41}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
Answer \( \left\langle - \frac{186}{41}, - \frac{31}{41}, - \frac{62}{41} \right\rangle \)
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: passqwen3.6:27b-mlx: pass — The solution correctly calculates the dot product and the squared magnitude, and applies the vector projection formula accurately.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly calculates the dot product and the squared magnitude, and applies the vector projection formula accurately.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The dot product calculation in step 1 is incorrect; it computes -24 + 5 + 12 = -7 (or similar) but states -31, missing the contribution from the z-components (u_z * v_z = -12). The correct dot product is -7, leading to a different projection.gpt-oss:20b: pass 2026-10-11
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.