Projections and orthogonality
Problem 9.428 · medium
Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -2, -1, -4 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -5, 2, 6 \right\rangle \).
- \[ -24 - 2 + 10 = -16 \]u·v.✓ Proved
- \[ 4 + 25 + 36 = 65 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}\frac{16}{13}\\- \frac{32}{65}\\- \frac{96}{65}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
Answer \( \left\langle \frac{16}{13}, - \frac{32}{65}, - \frac{96}{65} \right\rangle \)
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 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies the vector projection formula and the arithmetic is verified by the algebra system.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The dot product calculation is incorrect: u·v should be (-2)(-5) + (-1)(2) + (-4)(6) = 10 - 2 - 24 = -16, but the solution writes '-24 - 2 + 10 = -16' which implies the terms are -24, -2, and 10. While the sum is correct, the term '-2' corresponds to (-1)(2) which is correct, but the order and signs in the written sum are confusingly presented as if derived from a different calculation or just listed arbitrarily. More critically, the projection formula uses u·v = -16, so the scalar is -16/65. The final vector is (-16/65)*<-5, 2, 6> = <80/65, -32/65, -96/65> = <16/13, -32/65, -96/65>. The solution's final answer matches this. However, line 1 says '-24 - 2 + 10 = -16'. Let's re-verify the dot product terms. u=<-2,-1,-4>, v=<-5,2,6>. u1v1 = 10. u2v2 = -2. u3v3 = -24. Sum = 10 - 2 - 24 = -16. The line writes '-24 - 2 + 10'. This is arithmetically correct (-24-2+10 = -16). So the arithmetic is fine. Wait, looking closer at the prompt's provided solution text: '1. [equation, unchecked] -24 - 2 + 10 = -16 -- u·v.' This is correct. '2. [equation, unchecked] 4 + 25 + 36 = 65 -- ‖v‖².' This is correct. '3. [equation, unchecked] ... = ... -- proj = (u·v/‖v‖²) v.' The final vector is <16/13, -32/65, -96/65>. Let's check the scalar multiplication: (-16/65) * <-5, 2, 6> = <80/65, -32/65, -96/65>. 80/65 simplifies to 16/13. So the answer is correct. Why did I think it was wrong? I misread the sign of the first component in my head or something. Let's re-read carefully. u·v = -16. ||v||^2 = 65. Scalar = -16/65. v = <-5, 2, 6>. proj = <-80/65, -32/65, -96/65>? No. -16/65 * -5 = 80/65 = 16/13. -16/65 * 2 = -32/65. -16/65 * 6 = -96/65. The answer provided is <16/13, -32/65, -96/65>. This is correct. Is there any error? The prompt asks me to find what is wrong. If the solution is correct, I should say pass. Let me double check the dot product again. u = <-2, -1, -4>. v = <-5, 2, 6>. u·v = (-2)(-5) + (-1)(2) + (-4)(6) = 10 - 2 - 24 = -16. Correct. ||v||^2 = 25 + 4 + 36 = 65. Correct. Proj = (-16/65) <-5, 2, 6> = <80/65, -32/65, -96/65> = <16/13, -32/65, -96/65>. Correct. The solution is sound.gpt-oss:20b: pass 2026-10-08
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.