Dot product and angles
Problem 9.190 · easy
Let \( \displaystyle \mathbf{u} = \langle 4, -2, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle -5, 0, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
- \[ 4 \left(-5\right) - 4 \cdot 5 - 2 \cdot 0 = -40 \]Multiply componentwise and add.✓ Proved
- \[ 6 \]|u|.✓ Proved
- \[ 5 \sqrt{2} \]|v|.✓ Proved
- \[ - \frac{2 \sqrt{2}}{3} \]cos θ = u·v / (|u||v|).✓ Proved
- θ = arccos of that.
Answer \( \mathbf{u}\cdot\mathbf{v} = -40,\quad \theta = \operatorname{acos}{\left(- \frac{2 \sqrt{2}}{3} \right)} \approx 160.53^\circ \)
Lines: 4 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The dot product is computed incorrectly: it should be 4*(-5)+(-2)*0+(-4)*5 = -40, not 4*(-5)-4*5-2*0.qwen3.6:27b-mlx: pass — The solution correctly computes the dot product and magnitudes, applies the cosine formula for the angle, and provides the correct final answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly computes the dot product and magnitudes, applies the cosine formula for the angle, and provides the correct final answer.gpt-oss:20b: fail (error) 2026-10-03 — The dot product is computed incorrectly: it should be 4*(-5)+(-2)*0+(-4)*5 = -40, not 4*(-5)-4*5-2*0.qwen3.6:27b-mlx: inconclusive 2026-10-03 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitude of vector u is calculated incorrectly as 6; the correct magnitude is sqrt(36) = 6, wait, 4^2 + (-2)^2 + (-4)^2 = 16 + 4 + 16 = 36, sogpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.