Home›Calculus 3›Dot product and angles›Problem 9.166

Dot product and angles

Problem 9.166 · easy

Let \( \displaystyle \mathbf{u} = \langle 4, 3, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, 4, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-4\right) 5 + 4 \cdot 2 + 3 \cdot 4 = 0 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{41} \]
|u|.
\[ 3 \sqrt{5} \]
|v|.
\[ 0 \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = 0,\quad \theta = \frac{\pi}{2} \approx 90.00^\circ \)

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 solution fails to explicitly state the calculated value of the angle (pi/2 or 90 degrees), which is a required part of the problem statement. Additionally, the sentence 'theta = arccos of that' is vague and does not constitute a complete derivation or final answer.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the calculated value of the angle (pi/2 or 90 degrees), which is a required part of the problem statement. Additionally, the sentence 'theta = arccos of that' is vague and does not constitute a complete derivation or final answer.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to explicitly state the final answer for the angle, which is required by the problem statement. While the intermediate steps are correct, the solution ends with a procedural instruction rather than the result.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution fails to state the final answer explicitly. While the intermediate steps correctly identify that cos(theta) = 0, the final sentence 'theta = arccos of that' is not a complete answer and does not explicitly state that theta = pi/2 as required by the problem.
gpt-oss:20b: pass 2026-09-29

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.