Home›Calculus 3›Dot product and angles›Problem 9.5

Dot product and angles

Problem 9.5 · easy

Let \( \displaystyle \mathbf{u} = \langle -3, 4, 0 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, -2, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ 4 \left(-2\right) + 0 \cdot 2 - 3 \left(-1\right) = -5 \]
Multiply componentwise and add.✓ Proved
\[ 5 \]
|u|.
\[ 3 \]
|v|.
\[ - \frac{1}{3} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = -5,\quad \theta = \operatorname{acos}{\left(- \frac{1}{3} \right)} \approx 109.47^\circ \)

Lines: 1 proved, 4 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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: pass
qwen3.6:27b-mlx: fail (error) — The magnitude calculations are incorrect. |u| = sqrt((-3)^2 + 4^2) = 5, but the solution lists 3. |v| = sqrt((-1)^2 + (-2)^2 + 2^2) = 3, but the solution lists 5. The magnitudes were swapped, leading to an incorrect cosine value.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The magnitude calculations are incorrect. |u| = sqrt((-3)^2 + 4^2) = 5, but the solution lists 3. |v| = sqrt((-1)^2 + (-2)^2 + 2^2) = 3, but the solution lists 5. The magnitudes were swapped, leading to an incorrect cosine value.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The magnitude calculations are incorrect: |u| = 5, not 3, and |v| = 3, not 5. Consequently, the denominator |u||v| is 15, not 9, making cos(θ) = -1/5, not -1/3.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitudes |u| and |v| are calculated incorrectly (should be 5 and 3, not 5 and 3? Wait, |u|=5 is correct, |v|=sqrt(9)=3 is correct. Let me re-r
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 2026-10-13 with SymPy 1.14.0.