Home›Calculus 3›Dot product and angles›Problem 9.96

Dot product and angles

Problem 9.96 · easy

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

\[ \left(-1\right) \left(-1\right) - 1 \left(-5\right) + 5 \cdot 4 = 26 \]
Multiply componentwise and add.✓ Proved
\[ 3 \sqrt{3} \]
|u|.Reviewed
\[ \sqrt{42} \]
|v|.Reviewed
\[ \frac{13 \sqrt{14}}{63} \]
cos θ = u·v / (|u||v|).Reviewed
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = 26,\quad \theta = \operatorname{acos}{\left(\frac{13 \sqrt{14}}{63} \right)} \approx 39.46^\circ \)

Lines: 1 proved, 4 reviewed. 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	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, magnitudes, and cosine of the angle. The final step correctly identifies the angle as the arccosine of the computed value.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the dot product, magnitudes, and cosine of the angle. The final step correctly identifies the angle as the arccosine of the computed value.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The calculation of |u| in line 2 is incorrect. The magnitude of u = <-1, 5, -1> is sqrt(1 + 25 + 1) = sqrt(27) = 3*sqrt(3), which is correct, but th
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitude of vector u is calculated incorrectly as 3*sqrt(3) instead of sqrt(27) which simplifies to 3*sqrt(3), wait, sqrt(1+25+1) = sqrt(27) =
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.