Home›Calculus 3›Dot product and angles›Problem 9.49

Dot product and angles

Problem 9.49 · easy

Let \( \displaystyle \mathbf{u} = \langle -1, 2, 5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 0, -5, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ 2 \left(-5\right) - 1 \cdot 0 + 5 \cdot 4 = 10 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{30} \]
|u|.
\[ \sqrt{41} \]
|v|.
\[ \frac{\sqrt{1230}}{123} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = 10,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{1230}}{123} \right)} \approx 73.43^\circ \)

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 of the angle is computed incorrectly: it should be 10/√1230, not √1230/123. The solution’s final expression for cos θ is therefore wrong.
qwen3.6:27b-mlx: pass — The solution correctly calculates the dot product and magnitudes, applies the cosine formula accurately, and states the final angle in terms of arccos.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly calculates the dot product and magnitudes, applies the cosine formula accurately, and states the final angle in terms of arccos.
gpt-oss:20b: fail (error) 2026-09-28 — The cosine of the angle is computed incorrectly: it should be 10/√1230, not √1230/123. The solution’s final expression for cos θ is therefore wrong.
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution calculates the dot product as 10, but the correct value is -2 + 20 = 18. This error propagates to the final angle calculation.
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. The final answer matches the stated answer.
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.