Home›Calculus 3›Dot product and angles›Problem 9.3

Dot product and angles

Problem 9.3 · easy

Let \( \displaystyle \mathbf{u} = \langle 0, -4, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 3, -4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-4\right) 3 + 0 \cdot 5 - 4 \left(-4\right) = 4 \]
Multiply componentwise and add.✓ Proved
\[ 4 \sqrt{2} \]
|u|.Reviewed
\[ 5 \sqrt{2} \]
|v|.Reviewed
\[ 1 \cdot \frac{1}{10} = \frac{1}{10} \]
cos θ = u·v / (|u||v|).Reviewed
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = 4,\quad \theta = \operatorname{acos}{\left(\frac{1}{10} \right)} \approx 84.26^\circ \)

✓ Nihil obstat Lines: 1 proved, 4 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 as the arccosine of the result.

Every verdict on record (6)

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 as the arccosine of the result.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and identifies the angle via arccos.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle. No errors or misleading steps found.
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.