Home›Calculus 3›Dot product and angles›Problem 9.95

Dot product and angles

Problem 9.95 · easy

Let \( \displaystyle \mathbf{u} = \langle 0, -2, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -4, -4, -5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ 2 \left(-5\right) + 0 \left(-4\right) - 2 \left(-4\right) = -2 \]
Multiply componentwise and add.✓ Proved
\[ 2 \sqrt{2} \]
|u|.✓ Proved
\[ \sqrt{57} \]
|v|.✓ Proved
\[ \frac{\left(-1\right) \sqrt{114}}{114} = - \frac{\sqrt{114}}{114} \]
cos θ = u·v / (|u||v|).✓ Proved
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = -2,\quad \theta = \operatorname{acos}{\left(- \frac{\sqrt{114}}{114} \right)} \approx 95.37^\circ \)

Lines: 4 proved, 1 reviewed. 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	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: pass
qwen3.6:27b-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 2026-09-26 — 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 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The dot product calculation in step 1 is incorrect; it evaluates to 4 instead of the correct value 22. Consequently, the final angle is wrong.
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.