Home›Calculus 3›Dot product and angles›Problem 9.9

Dot product and angles

Problem 9.9 · easy

Let \( \displaystyle \mathbf{u} = \langle 2, 1, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 3, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ 2 \left(-1\right) + 1 \cdot 3 + 5 \cdot 5 = 26 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{30} \]
|u|.
\[ \sqrt{35} \]
|v|.
\[ \frac{13 \sqrt{42}}{105} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = 26,\quad \theta = \operatorname{acos}{\left(\frac{13 \sqrt{42}}{105} \right)} \approx 36.64^\circ \)

Lines: 1 proved, 4 not checked. 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
3	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Not checked	—	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: fail (error) — The cosine value is mis‑computed: 26/(√30·√35)=13/(5√42)=13√42/210, not 13√42/105. The angle formula is therefore incorrect.
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 in terms of arccos.

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 in terms of arccos.
gpt-oss:20b: fail (error) 2026-09-28 — The cosine value is mis‑computed: 26/(√30·√35)=13/(5√42)=13√42/210, not 13√42/105. The angle formula is therefore incorrect.
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and states the final angle using the inverse cosine function.
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 in terms of arccos as requested.
gpt-oss:20b: fail (error) 2026-09-26 — The cosine value is mis‑calculated: 26/(√30·√35)=26/(5√42)=13/(5√42)=13√42/210, not 13√42/105. The angle should be arccos(13√42/210).

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.