Dot product and angles
Problem 9.117 · easy
Let \( \displaystyle \mathbf{u} = \langle -1, 0, 3 \rangle \) and \( \displaystyle \mathbf{v} = \langle -3, -4, -1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
- \[ 3 \left(-1\right) + 0 \left(-4\right) - 1 \left(-3\right) = 0 \]Multiply componentwise and add.✓ Proved
- \[ \sqrt{10} \]|u|.✓ Proved
- \[ \sqrt{26} \]|v|.✓ Proved
- \[ 0 \]cos θ = u·v / (|u||v|).✓ Proved
- θ = arccos of that.
Answer \( \mathbf{u}\cdot\mathbf{v} = 0,\quad \theta = \frac{\pi}{2} \approx 90.00^\circ \)
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: fail (error) — The solution fails to explicitly state the final numerical answer for the angle (pi/2), stopping instead at the method description. Additionally, the notation in lines 2 and 3 is ambiguous and potentially confusing regarding which vector's magnitude is being calculated.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly state the final numerical answer for the angle (pi/2), stopping instead at the method description. Additionally, the notation in lines 2 and 3 is ambiguous and potentially confusing regarding which vector's magnitude is being calculated.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution calculates the cosine of the angle as 0 but fails to explicitly state that the angle is pi/2. It stops at the intermediate step 'theta = arccos of that' without providing the final numerical answer required by the problem.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.