Dot product and angles
Problem 9.170 · easy
Let \( \displaystyle \mathbf{u} = \langle 1, -5, -3 \rangle \) and \( \displaystyle \mathbf{v} = \langle 1, 3, 4 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.
- \[ \left(-5\right) 3 - 3 \cdot 4 + 1 \cdot 1 = -26 \]Multiply componentwise and add.✓ Proved
- \[ \sqrt{35} \]|u|.
- \[ \sqrt{26} \]|v|.
- \[ \frac{\left(-1\right) \sqrt{910}}{35} = - \frac{\sqrt{910}}{35} \]cos θ = u·v / (|u||v|).
- θ = arccos of that.
Answer \( \mathbf{u}\cdot\mathbf{v} = -26,\quad \theta = \operatorname{acos}{\left(- \frac{\sqrt{910}}{35} \right)} \approx 149.53^\circ \)
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 calculation is incorrect: α = -26/(√35·√26) = -26/√910, not -√910/35. This mis‑computes the angle.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The magnitude of vector v is calculated as sqrt(26) in line 3, but it should be sqrt(26) only if the components were different; actually |v| = sqrt(1^2 + 3^2 + 4^2) = sqrt(26). Wait, 1+9+16=26. So |v| is correct. Let me re-check |u|. |u| = sqrt(1^2 + (-5)^2 + (-3)^2) = sqrt(1+25+9) = sqrt(35). So |u| is correct. The dot product is 1*1 + (-5)*3 + (-3)*4 = 1 - 15 - 12 = -26. Correct. cos(theta) = -26 / (sqrt(35)*sqrt(26)) = -26 / sqrt(910). The solution writes -sqrt(910)/35. Let's check if -26/sqrt(910) equals -sqrt(910)/35. (-26/sqrt(910)) * (sqrt(910)/sqrt(910)) = -26*sqrt(910)/910. 910/26 = 35. So -26/910 = -1/35. Thus -sqrt(910)/35 is correct. The algebra is correct. Why did I think it was wrong? Let me re-read the prompt constraints. 'Find what is wrong'. Is there a subtle error? Line 4 says '-sqrt(910)/35 = -sqrt(910)/35 -- cos theta = ...'. This is an identity, not a derivation. But the prompt says equations are checked by CAS. The sentence 'theta = arccos of that' is vague but acceptable. Is there a domain error? No. Is the angle correct? acos(-sqrt(910)/35) approx acos(-0.857) approx 149 degrees. Seems correct. Wait, look at line 3 again. |v| = sqrt(26). Look at line 2. |u| = sqrt(35). Product is sqrt(910). Dot product -26. Cos theta = -26/sqrt(910). Rationalizing: -26*sqrt(910)/910 = -sqrt(910)/35. Everything seems correct. Let me look closer at the 'sentence' part. '5. [sentence] theta = arccos of that.' This is a bit informal but not 'wrong'. However, usually, one must specify the range of arccos or ensure the angle is in [0, pi]. Arccos returns values in [0, pi], which is the standard range for the angle between vectors. This is correct. Is there a typo in the problem statement vs solution? u = <1, -5, -3>, v = <1, 3, 4>. Dot product: 1 - 15 - 12 = -26. Correct. Magnitudes: sqrt(35), sqrt(26). Correct. Result: -sqrt(910)/35. Correct. Maybe the 'error' is that the solution doesn't explicitly state the formula for the angle? Line 4 has the comment 'cos theta = ...'. That's fine. Let me check the JSON output format. It requires 'notes'. If it's pass, notes empty. If I can't find an error, I should pass. But the prompt says 'Your job is to find what is wrong'. This implies there IS something wrong. Let me re-read carefully. 'sqrt(35) = sqrt(35) -- |u|'. 'sqrt(26) = sqrt(26) -- |v|'. These are tautologies marked as proved. The CAS checked them. The sentence 5 is 'theta = arccos of that'. This is extremely brief. Is it misleading? It doesn't define 'that'. But in context, it refers to line 4. Is there a precision issue? No. Let's look at the angle calculation. acos(-sqrt(910)/35). sqrt(910) approx 30.166. 30.166/35 approx 0.8618. acos(-0.8618) approx 149.5 degrees. The stated answer says 149.53. Let's check more precisely. 910^(0.5) = 30.166206. /35 = 0.8618916. acos(-0.8618916) in radians = 2.6099. In degrees = 149.537. Rounded to 149.54? Or 149.53? 149.537 rounds to 149.54. The stated answer says 149.53. This is a rounding error in the stated answer, but the solution text doesn't produce the decimal, it just says 'arccos of that'. The 'Stated answer' is separate from the 'Solution' lines. The solution lines end at 5. The solution does not claim the decimal value. So the solution itself is correct. Wait, is line 4's comment 'cos theta = u·v / (|u||v|)' applied correctly? Yes. Is there a sign error? No. I will mark it as pass because the mathematical steps are correct. The prompt asks to review the *solution*. The solution is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — [domain objection, downgraded to style] The magnitude of vector v is calculated as sqrt(26) in line 3, but it should be sqrt(26) only if the components were different; actually |v| = sqrt(1^2 + 3^2 + 4^2) = sqrt(26). Wait, 1+9+16=26. So |v| is correct. Let me re-check |u|. |u| = sqrt(1^2 + (-5)^2 + (-3)^2) = sqrt(1+25+9) = sqrt(35). So |u| is correct. The dot product is 1*1 + (-5)*3 + (-3)*4 = 1 - 15 - 12 = -26. Correct. cos(theta) = -26 / (sqrt(35)*sqrt(26)) = -26 / sqrt(910). The solution writes -sqrt(910)/35. Let's check if -26/sqrt(910) equals -sqrt(910)/35. (-26/sqrt(910)) * (sqrt(910)/sqrt(910)) = -26*sqrt(910)/910. 910/26 = 35. So -26/910 = -1/35. Thus -sqrt(910)/35 is correct. The algebra is correct. Why did I think it was wrong? Let me re-read the prompt constraints. 'Find what is wrong'. Is there a subtle error? Line 4 says '-sqrt(910)/35 = -sqrt(910)/35 -- cos theta = ...'. This is an identity, not a derivation. But the prompt says equations are checked by CAS. The sentence 'theta = arccos of that' is vague but acceptable. Is there a domain error? No. Is the angle correct? acos(-sqrt(910)/35) approx acos(-0.857) approx 149 degrees. Seems correct. Wait, look at line 3 again. |v| = sqrt(26). Look at line 2. |u| = sqrt(35). Product is sqrt(910). Dot product -26. Cos theta = -26/sqrt(910). Rationalizing: -26*sqrt(910)/910 = -sqrt(910)/35. Everything seems correct. Let me look closer at the 'sentence' part. '5. [sentence] theta = arccos of that.' This is a bit informal but not 'wrong'. However, usually, one must specify the range of arccos or ensure the angle is in [0, pi]. Arccos returns values in [0, pi], which is the standard range for the angle between vectors. This is correct. Is there a typo in the problem statement vs solution? u = <1, -5, -3>, v = <1, 3, 4>. Dot product: 1 - 15 - 12 = -26. Correct. Magnitudes: sqrt(35), sqrt(26). Correct. Result: -sqrt(910)/35. Correct. Maybe the 'error' is that the solution doesn't explicitly state the formula for the angle? Line 4 has the comment 'cos theta = ...'. That's fine. Let me check the JSON output format. It requires 'notes'. If it's pass, notes empty. If I can't find an error, I should pass. But the prompt says 'Your job is to find what is wrong'. This implies there IS something wrong. Let me re-read carefully. 'sqrt(35) = sqrt(35) -- |u|'. 'sqrt(26) = sqrt(26) -- |v|'. These are tautologies marked as proved. The CAS checked them. The sentence 5 is 'theta = arccos of that'. This is extremely brief. Is it misleading? It doesn't define 'that'. But in context, it refers to line 4. Is there a precision issue? No. Let's look at the angle calculation. acos(-sqrt(910)/35). sqrt(910) approx 30.166. 30.166/35 approx 0.8618. acos(-0.8618) approx 149.5 degrees. The stated answer says 149.53. Let's check more precisely. 910^(0.5) = 30.166206. /35 = 0.8618916. acos(-0.8618916) in radians = 2.6099. In degrees = 149.537. Rounded to 149.54? Or 149.53? 149.537 rounds to 149.54. The stated answer says 149.53. This is a rounding error in the stated answer, but the solution text doesn't produce the decimal, it just says 'arccos of that'. The 'Stated answer' is separate from the 'Solution' lines. The solution lines end at 5. The solution does not claim the decimal value. So the solution itself is correct. Wait, is line 4's comment 'cos theta = u·v / (|u||v|)' applied correctly? Yes. Is there a sign error? No. I will mark it as pass because the mathematical steps are correct. The prompt asks to review the *solution*. The solution is correct.gpt-oss:20b: fail (error) 2026-10-04 — The cosine calculation is incorrect: α = -26/(√35·√26) = -26/√910, not -√910/35. This mis‑computes the angle.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The calculation of |u| is incorrect; sqrt(1^2 + (-5)^2 + (-3)^2) is sqrt(35), but the solution lists it as sqrt(35) while the subsequent cosine calculation implies a denominator of 35 (sqrt(35)*sqrt(26) = sqrt(910)). Wait, let's re-read carefully. Line 2 says sqrt(35). Line 3 says sqrt(26). Line 4 says -sqrt(910)/35. sqrt(35)*sqrt(26) = sqrt(910). So the denominator should be sqrt(910). The solution writes -sqrt(910)/35. This implies the denominator was calculated as 35, not sqrt(910). 35 is 5*7. sqrt(35)*sqrt(26) is not 35. The algebra in line 4 is marked 'proved' by the CAS, but the text description 'cos theta = u.v / (|u||v|)' combined with the result -sqrt(910)/35 suggests a mismatch. Let's check the CAS result. u.v = -26. |u| = sqrt(35). |v| = sqrt(26). cos(theta) = -26 / (sqrt(35)*sqrt(26)) = -26 / sqrt(910). Rationalizing: -26*sqrt(910)/910 = -sqrt(910)/35. Ah, -26/910 = -1/35. So -sqrt(910)/35 is actually correct. My initial check was hasty. -26/sqrt(910) = -26*sqrt(910)/910. 910/26 = 35. So yes, -sqrt(910)/35 is correct. The setup and calculations are correct. The sentence is brief but correct.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (error) 2026-09-29 — The magnitude of vector v is calculated as sqrt(26), but the actual magnitude is sqrt(1^2 + 3^2 + 4^2) = sqrt(26). Wait, 1+9+16=26. So |v| is correct. Let me re-check |u|. u = <1, -5, -3>. |u|^2 = 1 + 25 + 9 = 35. So |u| = sqrt(35). Correct. Dot product: 1*1 + (-5)*3 + (-3)*4 = 1 - 15 - 12 = -26. Correct. cos(theta) = -26 / (sqrt(35)*sqrt(26)) = -26 / sqrt(910). The solution simplifies this to -sqrt(910)/35. Let's check: -26 / sqrt(910) = -26*sqrt(910)/910 = -sqrt(910)/35. This is correct. The final answer is acos(-sqrt(910)/35). The solution seems correct. Why did I think it was wrong? Let me re-read the prompt. 'Find u . v and the angle between them.' The solution calculates the dot product (-26) and the angle. The stated answer is just the angle. The solution steps are correct. Is there a subtle error? Step 1 says '-5*3 - 3*4 + 1*1'. This is -15 - 12 + 1 = -26. Correct. Step 2: sqrt(35). Correct. Step 3: sqrt(26). Correct. Step 4: -sqrt(910)/35. Correct. Step 5: theta = arccos of that. Correct. The verdict should be pass. Wait, looking closer at the provided 'Stated answer': acos(-sqrt(910)/35). The solution derives this. Is there any issue with the range of arccos? No, arccos returns values in [0, pi], which is the standard range for the angle between vectors. The solution is sound.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.