Home›Calculus 3›Dot product and angles›Problem 9.48

Dot product and angles

Problem 9.48 · easy

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

\[ 2 \left(-4\right) - 1 \cdot 2 + 5 \cdot 4 = 10 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{30} \]
|u|.Reviewed
\[ 6 \]
|v|.Reviewed
\[ \frac{\sqrt{30}}{18} \]
cos θ = u·v / (|u||v|).Reviewed
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = 10,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{30}}{18} \right)} \approx 72.28^\circ \)

✓ 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 for the angle between vectors, and states the final result accurately.

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 for the angle between vectors, and states the final result accurately.
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 correctly identifies the angle as the arccosine of the result.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly calculates the dot product and magnitudes, applies the cosine formula for the angle between vectors, and states the final result. The steps are logically sound and algebraically correct.
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.