Dot product and angles
Problem 9.146 · easy
Let \( \displaystyle \mathbf{u} = \langle 4, 0, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, 4, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
- \[ 0 \cdot 4 + 4 \cdot 3 - 5 \left(-5\right) = 37 \]Multiply componentwise and add.✓ Proved
- \[ \sqrt{41} \]|u|.✓ Proved
- \[ 5 \sqrt{2} \]|v|.✓ Proved
- \[ \frac{37 \sqrt{82}}{410} \]cos θ = u·v / (|u||v|).✓ Proved
- θ = arccos of that.Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = 37,\quad \theta = \operatorname{acos}{\left(\frac{37 \sqrt{82}}{410} \right)} \approx 35.20^\circ \)
Lines: 4 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the law of cosines on the triangle u, v, u − v gives the same angle |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle. No errors found.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle. No errors found.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitude of vector v is calculated incorrectly in step 3. |v| = sqrt(3^2 + 4^2 + (-5)^2) = sqrt(9 + 16 + 25) = sqrt(50) = 5*sqrt(2), but the sogpt-oss:20b: pass 2026-09-28
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/dot_product_angle, checked 2026-09-28 with SymPy 1.14.0.