Home›Calculus 3›Dot product and angles›Problem 9.115

Dot product and angles

Problem 9.115 · easy

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

\[ 5 \left(-4\right) - 5 \cdot 4 - 4 \cdot 5 = -60 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{66} \]
|u|.✓ Proved
\[ \sqrt{57} \]
|v|.✓ Proved
\[ - \frac{10 \sqrt{418}}{209} \]
cos θ = u·v / (|u||v|).✓ Proved
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = -60,\quad \theta = \operatorname{acos}{\left(- \frac{10 \sqrt{418}}{209} \right)} \approx 168.02^\circ \)

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: fail (error) — The dot product calculation in line 1 is incorrect; 5*(-4) + (-5)*4 + (-4)*5 equals -50, not -60. This error propagates to the final angle.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The dot product calculation in line 1 is incorrect; 5*(-4) + (-5)*4 + (-4)*5 equals -50, not -60. This error propagates to the final angle.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The dot product calculation in step 1 is incorrect; it sums to -60 instead of the correct value of 0. Consequently, the angle is incorrectly calculated as non-zero instead of pi/2.
gpt-oss:20b: pass 2026-09-27

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 2026-09-27 with SymPy 1.14.0.