Dot product and angles
Problem 9.118 · easy
Let \( \displaystyle \mathbf{u} = \langle 5, 0, 4 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, -5, 1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
- \[ 5 \left(-4\right) + 0 \left(-5\right) + 4 \cdot 1 = -16 \]Multiply componentwise and add.✓ Proved
- \[ \sqrt{41} \]|u|.✓ Proved
- \[ \sqrt{42} \]|v|.✓ Proved
- \[ - \frac{8 \sqrt{1722}}{861} \]cos θ = u·v / (|u||v|).✓ Proved
- θ = arccos of that.Reviewed
Answer \( \mathbf{u}\cdot\mathbf{v} = -16,\quad \theta = \operatorname{acos}{\left(- \frac{8 \sqrt{1722}}{861} \right)} \approx 112.68^\circ \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 identifies the angle via arccos. The final answer matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and identifies the angle via arccos. The final answer matches the stated answer.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and identifies the angle via arccos. The final expression matches the stated answer.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.