Home›Calculus 3›Dot product and angles›Problem 9.167

Dot product and angles

Problem 9.167 · easy

Let \( \displaystyle \mathbf{u} = \langle -1, -3, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 1, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-3\right) 1 - 1 \left(-1\right) + 2 \cdot 2 = 2 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{14} \]
|u|.
\[ \sqrt{6} \]
|v|.
\[ \frac{\sqrt{21}}{21} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = 2,\quad \theta = \operatorname{acos}{\left(\frac{\sqrt{21}}{21} \right)} \approx 77.40^\circ \)

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 dot product calculation in line 1 is incorrect; it computes 3 instead of the correct value 2 (since (-1)(-1) + (-3)(1) + (2)(2) = 1 - 3 + 4 = 2). Although the final stated answer claims the dot product is 2, the derivation shown is wrong, and the angle calculation relies on the incorrect intermediate values or inconsistent logic.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The dot product calculation in line 1 is incorrect; it computes 3 instead of the correct value 2 (since (-1)(-1) + (-3)(1) + (2)(2) = 1 - 3 + 4 = 2). Although the final stated answer claims the dot product is 2, the derivation shown is wrong, and the angle calculation relies on the incorrect intermediate values or inconsistent logic.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The dot product calculation in step 1 is incorrect; it computes 6 instead of the correct value 5. This leads to an incorrect angle.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — The dot product calculation in line 1 is incorrect; it computes 3 instead of the correct value 7. Consequently, the magnitudes and final angle are also wrong.
gpt-oss:20b: pass 2026-09-29

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.