Home›Calculus 3›Dot product and angles›Problem 9.11

Dot product and angles

Problem 9.11 · easy

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

\[ \left(-2\right) 4 - 2 \cdot 1 + 1 \cdot 5 = -5 \]
Multiply componentwise and add.✓ Proved
\[ 3 \]
|u|.
\[ \sqrt{42} \]
|v|.
\[ - \frac{5 \sqrt{42}}{126} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = -5,\quad \theta = \operatorname{acos}{\left(- \frac{5 \sqrt{42}}{126} \right)} \approx 104.90^\circ \)

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: pass
qwen3.6:27b-mlx: fail (error) — The calculation for |u| is incorrect; |u| should be sqrt(9) = 3, but the solution lists 3 and 3 as if they were the components or an intermediate step without squaring, and more critically, the final cosine value is wrong because it divides by 3*sqrt(42) but the numerator -5 is correct, yet the simplification -5/(3*sqrt(42)) = -5*sqrt(42)/42, not /126. The denominator 126 implies a multiplication by 3 somewhere that shouldn't be there or a calculation error in the magnitude product.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The calculation for |u| is incorrect; |u| should be sqrt(9) = 3, but the solution lists 3 and 3 as if they were the components or an intermediate step without squaring, and more critically, the final cosine value is wrong because it divides by 3*sqrt(42) but the numerator -5 is correct, yet the simplification -5/(3*sqrt(42)) = -5*sqrt(42)/42, not /126. The denominator 126 implies a multiplication by 3 somewhere that shouldn't be there or a calculation error in the magnitude product.
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. The final answer matches the stated answer.
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 identifies the angle via arccos.
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.