Projections and orthogonality
Problem 9.582 · medium
Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -5, 5, 2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -1, -5, 6 \right\rangle \).
- \[ -25 + 5 + 12 = -8 \]u·v.✓ Proved
- \[ 1 + 25 + 36 = 62 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}\frac{4}{31}\\\frac{20}{31}\\- \frac{24}{31}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
Answer \( \left\langle \frac{4}{31}, \frac{20}{31}, - \frac{24}{31} \right\rangle \)
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 applies the vector projection formula and the arithmetic is verified by the algebra system.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly applies the vector projection formula and the arithmetic is verified by the algebra system.qwen3.6:27b-mlx: fail (error) 2026-10-13 — The dot product calculation in line 1 is incorrect: (-5)(-1) + (5)(-5) + (2)(6) = 5 - 25 + 12 = -8, but the line states -25 + 5 + 12 = -8, implying the first term was calculated as -25 instead of 5. Although the final result is numerically correct because the error in the dot product (-8 vs -8) is coincidental (wait, -8 is correct, let me re-read carefully). Re-evaluating line 1: u = <-5, 5, 2>, v = <-1, -5, 6> u · v = (-5)(-1) + (5)(-5) + (2)(6) = 5 - 25 + 12 = -8. The line says: -25 + 5 + 12 = -8. This implies the terms are: (-5)(5)? No. It looks like the author calculated (-5)(5) = -25 for the first component? Or perhaps they swapped signs? Actually, (-5)*(-1) = 5. The line writes -25. This is an arithmetic error in the intermediate step description, even if the sum -8 is correct. Wait, let's look closer. Term 1: -5 * -1 = 5. Line has -25. Term 2: 5 * -5 = -25. Line has 5. Term 3: 2 * 6 = 12. Line has 12. Sum in line: -25 + 5 + 12 = -8. Actual sum: 5 - 25 + 12 = -8. The line contains swapped values for the first two terms (-25 and 5 instead of 5 and -25). While the final sum is correct, the breakdown is factually wrong. This is misleading/error-prone for a student checking their work. However, usually 'error' is reserved for steps that lead to a wrong answer or violate logic. Here the logic holds and the answer is right. Is it 'misleading'? Yes. A student might think (-5)(-1) = -25. Let's check the projection formula application. proj_v u = (u·v / ||v||^2) v. u·v = -8. ||v||^2 = 62. Scalar = -8/62 = -4/31. Vector = (-4/31) * <-1, -5, 6> = <4/31, 20/31, -24/31>. The final answer is correct. The issue is strictly in line 1's arithmetic breakdown. It claims -25 + 5 + 12. The actual terms are 5, -25, 12. The line has swapped the first two terms' values. This is a factual error in the explanation of the dot product. Verdict: fail. Severity: misleading (or error, depending on strictness. It's an arithmetic error in the display). I will call it error because the equation presented is false: the terms listed do not correspond to the dot product components.gpt-oss:20b: pass 2026-10-13
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 2026-10-13 with SymPy 1.14.0.