Dot product and angles
Problem 9.98 · easy
Let \( \displaystyle \mathbf{u} = \langle 2, -3, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 4, 0, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
- \[ \left(-3\right) 0 + 2 \cdot 4 - 3 \left(-5\right) = 23 \]Multiply componentwise and add.✓ Proved
- \[ \sqrt{22} \]|u|.✓ Proved
- \[ \sqrt{41} \]|v|.✓ Proved
- \[ \frac{23 \sqrt{902}}{902} \]cos θ = u·v / (|u||v|).✓ Proved
- θ = arccos of that.
Answer \( \mathbf{u}\cdot\mathbf{v} = 23,\quad \theta = \operatorname{acos}{\left(\frac{23 \sqrt{902}}{902} \right)} \approx 40.02^\circ \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitude of vector u is calculated as sqrt(22) in line 2, but |u| = sqrt(2^2 + (-3)^2 + (-3)^2) = sqrt(22) is incorrect; it should be sqrt(4 +
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitude of vector u is calculated as sqrt(22) in line 2, but |u| = sqrt(2^2 + (-3)^2 + (-3)^2) = sqrt(22) is incorrect; it should be sqrt(4 +gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle in terms of arccos.gpt-oss:20b: pass 2026-09-26
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 2026-09-26 with SymPy 1.14.0.