Projections and orthogonality
Problem 9.262 · hard
Write \( \displaystyle \mathbf u = \left\langle -5, -5, -5 \right\rangle \) as the sum of a vector parallel to \( \displaystyle \mathbf v = \left\langle -1, -4, 3 \right\rangle \) and a vector orthogonal to it.
- \[ -15 + 5 + 20 = 10 \]u·v.✓ Proved
- \[ 1 + 9 + 16 = 26 \]‖v‖².✓ Proved
- \[ \left[\begin{matrix}- \frac{5}{13}\\- \frac{20}{13}\\\frac{15}{13}\end{matrix}\right] \]proj = (u·v/‖v‖²) v.✓ Proved
- \[ \left[\begin{matrix}- \frac{60}{13}\\- \frac{45}{13}\\- \frac{80}{13}\end{matrix}\right] \]The orthogonal part is u − proj.✓ Proved
- \[ 0 \]It is orthogonal to v.✓ Proved
Answer \( \left\langle - \frac{5}{13}, - \frac{20}{13}, \frac{15}{13} \right\rangle + \left\langle - \frac{60}{13}, - \frac{45}{13}, - \frac{80}{13} \right\rangle \)
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 |
| 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 computes the projection and orthogonal components, and verifies orthogonality. The steps are logically sound and algebraically correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly computes the projection and orthogonal components, and verifies orthogonality. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The dot product calculation in line 1 is incorrect: u·v = (-5)(-1) + (-5)(-4) + (-5)(3) = 5 + 20 - 15 = 10, but the line states '-15 + 5 + 20 = 10', which implies the terms are ordered or signed incorrectly relative to the vector components, although the sum is coincidentally correct. However, the main error is in the projection calculation. The projection of u onto v is (u·v / ||v||^2) * v. u·v = 10, ||v||^2 = 26. So proj = (10/26)v = (5/13)v = < -5/13, -20/13, 15/13 >. This matches line 3. The orthogonal part is u - proj = < -5, -5, -5 > - < -5/13, -20/13, 15/13 > = < -60/13, -45/13, -80/13 >. This matches line 4. Wait, let me re-read the prompt's provided solution lines. Line 1 says '-15 + 5 + 20 = 10'. The components of u are -5, -5, -5 and v are -1, -4, 3. The products are 5, 20, -15. The sum is 10. The line writes '-15 + 5 + 20'. This is just a permutation of the terms, so the arithmetic is correct. The projection vector is correct. The orthogonal vector is correct. The orthogonality check is correct. Why would this fail? Let me check the arithmetic again. u = <-5, -5, -5>, v = <-1, -4, 3>. u·v = 5 + 20 - 15 = 10. ||v||^2 = 1 + 16 + 9 = 26. proj_v u = (10/26) <-1, -4, 3> = (5/13) <-1, -4, 3> = <-5/13, -20/13, 15/13>. u - proj = <-5 + 5/13, -5 + 20/13, -5 - 15/13> = <-60/13, -45/13, -80/13>. The solution is mathematically correct. Is there a style issue? The prompt asks to find what is wrong. If it's correct, I should say pass. But wait, look at line 1 again. '-15 + 5 + 20 = 10'. The terms correspond to z, x, y components? Usually one writes them in order. It's not an error, just unordered. Let me look closer at the vectors. u = <-5, -5, -5>. v = <-1, -4, 3>. The solution calculates the projection correctly. The orthogonal component correctly. The check correctly. It seems sound. However, often these 'adversarial' reviews catch subtle sign errors. Let's re-verify the orthogonal part calculation in line 4. u - proj = <-5, -5, -5> - <-5/13, -20/13, 15/13> = <-65/13 + 5/13, -65/13 + 20/13, -65/13 - 15/13> = <-60/13, -45/13, -80/13>. Correct. Line 5 check: dot product of orthogonal part and v: (-60/13)(-1) + (-45/13)(-4) + (-80/13)(3) = 60/13 + 180/13 - 240/13 = (240 - 240)/13 = 0. Correct. The solution is correct. I will mark it as pass.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.