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Dot product and angles

Problem 9.145 · easy

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

\[ 1 \left(-3\right) + 1 \left(-3\right) + 0 \left(-1\right) = -6 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{2} \]
|u|.✓ Proved
\[ \sqrt{19} \]
|v|.✓ Proved
\[ - \frac{3 \sqrt{38}}{19} \]
cos θ = u·v / (|u||v|).✓ Proved
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = -6,\quad \theta = \operatorname{acos}{\left(- \frac{3 \sqrt{38}}{19} \right)} \approx 166.74^\circ \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle in terms of arccos.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle in terms of arccos.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and identifies the angle using arccos. The final answer matches the stated answer.
gpt-oss:20b: pass 2026-09-28

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 2026-09-28 with SymPy 1.14.0.