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|.✓ Proved
\[ \sqrt{42} \]
|v|.✓ Proved
\[ \frac{13 \sqrt{14}}{63} \]
cos θ = u·v / (|u||v|).✓ Proved
θ = arccos of that.

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

Lines: 4 proved, 1 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.

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	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: inconclusive — 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

Every verdict on record (4)

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 2026-09-26 with SymPy 1.14.0.