Home›Calculus 3›Dot product and angles›Problem 9.92

Dot product and angles

Problem 9.92 · easy

Let \( \displaystyle \mathbf{u} = \langle 0, 3, -2 \rangle \) and \( \displaystyle \mathbf{v} = \langle 5, 0, 2 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-2\right) 2 + 0 \cdot 5 + 3 \cdot 0 = -4 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{13} \]
|u|.✓ Proved
\[ \sqrt{29} \]
|v|.✓ Proved
\[ - \frac{4 \sqrt{377}}{377} \]
cos θ = u·v / (|u||v|).✓ Proved
θ = arccos of that.
Reviewed

Answer \( \mathbf{u}\cdot\mathbf{v} = -4,\quad \theta = \operatorname{acos}{\left(- \frac{4 \sqrt{377}}{377} \right)} \approx 101.89^\circ \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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	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 and magnitudes, applies the cosine formula accurately, and correctly identifies the angle as the arccosine of the result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the dot product and magnitudes, applies the cosine formula accurately, and correctly identifies the angle as the arccosine of the result.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (style) 2026-09-26 — [domain objection, downgraded to style] The magnitude calculations in lines 2 and 3 are incorrect. |u| should be sqrt(13) but is listed as sqrt(13) (wait, 0^2+3^2+(-2)^2 = 13, so sqrt(13) is correct for u). |v| should be sqrt(29) (5^2+0^2+2^2 = 29), but line 3 lists sqrt(29). Let me re-read carefully. Line 2 says sqrt(13), sqrt(13). Line 3 says sqrt(29), sqrt(29). The dot product is -4. The denominator is sqrt(13)*sqrt(29) = sqrt(377). The cosine is -4/sqrt(377). Line 4 writes -4*sqrt(377)/377, which is the rationalized form of -4/sqrt(377). This is algebraically correct. However, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Lines 1-4 are marked [equation, unchecked]. The review instructions say 'do not re-check the algebra of a line marked proved'. But these are marked 'unchecked'. The instructions say 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. These are NOT marked proved. They are marked unchecked. This implies I should check them? Or does 'unchecked' mean the CAS didn't check them? The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. It doesn't explicitly say I must check unchecked equations, but usually 'unchecked' implies they haven't been verified. However, the core task is to review the *sentences* and the *logic*. Let's look at the logic. The setup is correct. The formulas are correct. The arithmetic in the equations, if unchecked, might be wrong. Let's double check the arithmetic. u=<0,3,-2>, v=<5,0,2>. u.v = 0*5 + 3*0 + (-2)*2 = -4. Correct. |u| = sqrt(0+9+4) = sqrt(13). Correct. |v| = sqrt(25+0+4) = sqrt(29). Correct. cos(theta) = -4 / (sqrt(13)sqrt(29)) = -4/sqrt(377). Rationalized: -4sqrt(377)/377. Correct. The sentence 'theta = arccos of that' is correct. Why would this fail? Maybe the 'unchecked' tag is a trap? Or maybe I missed a subtle error. Let's look at the stated answer: acos(-4*sqrt(377)/377). This matches. Is there a domain issue? No. Is the angle between vectors defined as [0, pi]? Yes, arccos returns [0, pi]. Everything seems correct. Wait, look at line 2 and 3 again. Line 2: `sqrt(13), sqrt(13)`. Line 3: `sqrt(29), sqrt(29)`. The format seems to be `expression, result`. The result is correct. The sentence is line 5. It is correct. Is there a style issue? 'arccos of that' is informal but not misleading. Let me re-read the prompt constraints. 'Check that... the setup models the problem... each theorem... each conclusion follows... nothing would teach a student something false.' Everything seems fine. Is it possible the verdict should be pass? Let me check the vector components again. u = <0, 3, -2>. v = <5, 0, 2>. Dot product: -4. Norms: sqrt(13), sqrt(29). Cosine: -4/sqrt(377). Angle: arccos(-4/sqrt(377)). The solution provides the rationalized form. This is standard. I see no error. I will mark it pass.
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.