Home›Calculus 3›Dot product and angles›Problem 9.139

Dot product and angles

Problem 9.139 · easy

Let \( \displaystyle \mathbf{u} = \langle -2, 4, 2 \rangle \) and \( \displaystyle \mathbf{v} = \langle -1, 3, 5 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-2\right) \left(-1\right) + 2 \cdot 5 + 4 \cdot 3 = 24 \]
Multiply componentwise and add.✓ Proved
\[ 2 \sqrt{6} \]
|u|.
\[ \sqrt{35} \]
|v|.
\[ \frac{2 \sqrt{210}}{35} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = 24,\quad \theta = \operatorname{acos}{\left(\frac{2 \sqrt{210}}{35} \right)} \approx 34.10^\circ \)

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 calculation for |u| is incorrect; |u| should be sqrt(24) = 2*sqrt(6), but the solution implies |u| = 2*sqrt(6) while the final cosine calculation uses a denominator consistent with |u| = sqrt(24) and |v| = sqrt(35) resulting in 24/sqrt(840) = 2*sqrt(210)/35. Wait, let's re-verify. u=<-2,4,2>, |u|^2 = 4+16+4=24, |u|=2sqrt(6). v=<-1,3,5>, |v|^2 = 1+9+25=35, |v|=sqrt(35). |u||v| = 2sqrt(6)sqrt(35) = 2sqrt(210). u.v = 24. cos(theta) = 24 / (2sqrt(210)) = 12/sqrt(210). The solution claims cos(theta) = 2*sqrt(210)/35. Let's check if 12/sqrt(210) equals 2*sqrt(210)/35. 12/sqrt(210) = 12*sqrt(210)/210 = 2*sqrt(210)/35. Yes, the algebra is correct. However, the sentence 'theta = arccos of that' is vague but acceptable. Is there an error? The problem asks for the angle. The stated answer gives approx 34.10 degrees. Let's check acos(2*sqrt(210)/35). 2*sqrt(210)/35 approx 2*14.49/35 approx 0.828. acos(0.828) approx 34.1 degrees. The solution steps are algebraically correct. Why would it fail? Ah, look at step 2 and 3. They are just identity equations. Step 4 is an identity equation. The 'sentence' in step 5 is the only non-trivial part. It says 'theta = arccos of that'. This is correct. Is there a hidden error? The prompt says 'Find u.v and the angle'. The solution finds u.v=24. It finds cos(theta). It states theta = arccos(...). This seems correct. Let me re-read the adversarial instructions. 'Check that... each conclusion follows from the lines before it'. Step 5 follows from 4. Step 4 follows from 1,2,3. The algebra is proved. The sentences are minimal. Is '2*sqrt(6)' in step 2 actually |u|? Yes. Is 'sqrt(35)' in step 3 actually |v|? Yes. Is the division in step 4 correct? 24 / (2*sqrt(6)*sqrt(35)) = 12/sqrt(210) = 12*sqrt(210)/210 = 2*sqrt(210)/35. Yes. The solution is correct. Why did I think it might fail? Maybe the angle unit? The stated answer uses degrees. The solution doesn't specify units in the sentence, but the stated answer does. The solution itself is just the derivation. It seems sound. Wait, looking closely at the provided solution text: '2. [equation, proved] 2*sqrt(6) = 2*sqrt(6) -- |u|.' This is a tautology. It doesn't show the calculation of |u|. But the prompt says 'equation... checked by CAS'. So we assume the value is correct. The sentence 'theta = arccos of that' is technically correct but 'that' refers to the value in step 4. This is acceptable. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The calculation for |u| is incorrect; |u| should be sqrt(24) = 2*sqrt(6), but the solution implies |u| = 2*sqrt(6) while the final cosine calculation uses a denominator consistent with |u| = sqrt(24) and |v| = sqrt(35) resulting in 24/sqrt(840) = 2*sqrt(210)/35. Wait, let's re-verify. u=<-2,4,2>, |u|^2 = 4+16+4=24, |u|=2sqrt(6). v=<-1,3,5>, |v|^2 = 1+9+25=35, |v|=sqrt(35). |u||v| = 2sqrt(6)sqrt(35) = 2sqrt(210). u.v = 24. cos(theta) = 24 / (2sqrt(210)) = 12/sqrt(210). The solution claims cos(theta) = 2*sqrt(210)/35. Let's check if 12/sqrt(210) equals 2*sqrt(210)/35. 12/sqrt(210) = 12*sqrt(210)/210 = 2*sqrt(210)/35. Yes, the algebra is correct. However, the sentence 'theta = arccos of that' is vague but acceptable. Is there an error? The problem asks for the angle. The stated answer gives approx 34.10 degrees. Let's check acos(2*sqrt(210)/35). 2*sqrt(210)/35 approx 2*14.49/35 approx 0.828. acos(0.828) approx 34.1 degrees. The solution steps are algebraically correct. Why would it fail? Ah, look at step 2 and 3. They are just identity equations. Step 4 is an identity equation. The 'sentence' in step 5 is the only non-trivial part. It says 'theta = arccos of that'. This is correct. Is there a hidden error? The prompt says 'Find u.v and the angle'. The solution finds u.v=24. It finds cos(theta). It states theta = arccos(...). This seems correct. Let me re-read the adversarial instructions. 'Check that... each conclusion follows from the lines before it'. Step 5 follows from 4. Step 4 follows from 1,2,3. The algebra is proved. The sentences are minimal. Is '2*sqrt(6)' in step 2 actually |u|? Yes. Is 'sqrt(35)' in step 3 actually |v|? Yes. Is the division in step 4 correct? 24 / (2*sqrt(6)*sqrt(35)) = 12/sqrt(210) = 12*sqrt(210)/210 = 2*sqrt(210)/35. Yes. The solution is correct. Why did I think it might fail? Maybe the angle unit? The stated answer uses degrees. The solution doesn't specify units in the sentence, but the stated answer does. The solution itself is just the derivation. It seems sound. Wait, looking closely at the provided solution text: '2. [equation, proved] 2*sqrt(6) = 2*sqrt(6) -- |u|.' This is a tautology. It doesn't show the calculation of |u|. But the prompt says 'equation... checked by CAS'. So we assume the value is correct. The sentence 'theta = arccos of that' is technically correct but 'that' refers to the value in step 4. This is acceptable. I will mark it as pass.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The dot product calculation in line 1 is incorrect; it omits the x-component product (-2)*(-1) = 2, resulting in 24 instead of the correct value 26. This leads to an incorrect angle.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The dot product calculation in line 1 is incorrect; it omits the y-component product (4*3 is listed but -2*-1 is 2, 4*3 is 12, 2*5 is 10, sum is 24?
gpt-oss:20b: pass 2026-09-28

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.