Home›Calculus 3›Dot product and angles›Problem 9.93

Dot product and angles

Problem 9.93 · easy

Let \( \displaystyle \mathbf{u} = \langle -5, 0, -4 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -1, 0 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-5\right) 2 + 0 \left(-1\right) - 4 \cdot 0 = -10 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{41} \]
|u|.✓ Proved
\[ \sqrt{5} \]
|v|.✓ Proved
\[ - \frac{2 \sqrt{205}}{41} \]
cos θ = u·v / (|u||v|).✓ Proved
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = -10,\quad \theta = \operatorname{acos}{\left(- \frac{2 \sqrt{205}}{41} \right)} \approx 134.30^\circ \)

✓ 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 correctly identifies the angle via arccos.

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 correctly identifies the angle via arccos.
gpt-oss:20b: pass 2026-09-26
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

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.