Home›Calculus 3›Dot product and angles›Problem 9.53

Dot product and angles

Problem 9.53 · easy

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

\[ \left(-3\right) 5 + 5 \left(-3\right) + 5 \cdot 1 = -25 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{59} \]
|u|.
\[ \sqrt{35} \]
|v|.
\[ - \frac{5 \sqrt{2065}}{413} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = -25,\quad \theta = \operatorname{acos}{\left(- \frac{5 \sqrt{2065}}{413} \right)} \approx 123.38^\circ \)

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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitude calculations are incorrect: |u| = sqrt(9+25+25) = sqrt(59) is correct, but |v| = sqrt(25+1+9) = sqrt(35) is incorrect (should be sqrt(

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The magnitude calculations are incorrect: |u| = sqrt(9+25+25) = sqrt(59) is correct, but |v| = sqrt(25+1+9) = sqrt(35) is incorrect (should be sqrt(
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The dot product calculation in step 1 is incorrect: (-3)(5) + (5)(1) + (5)(-3) = -15 + 5 - 15 = -25 is wrong, the correct sum is -25? Wait. -15 + 5 - 15 = -25. Let me re-read the vectors. u = <-3, 5, 5>, v = <5, 1, -3>. u·v = -15 + 5 - 15 = -25. The equation says -25. Okay, the dot product is correct. Let's check magnitudes. |u|^2 = 9 + 25 + 25 = 59. |v|^2 = 25 + 1 + 9 = 35. So |u| = sqrt(59), |v| = sqrt(35). Step 2 and 3 are correct. Step 4 calculates cos(theta) = -25 / (sqrt(59)*sqrt(35)) = -25 / sqrt(2065). The solution simplifies this to -5*sqrt(2065)/413. Let's check: -25/sqrt(2065) = -25*sqrt(2065)/2065. 2065 / 5 = 413. So -5*sqrt(2065)/413. This is correct. Step 5 states theta = arccos of that. The stated answer is acos(-5*sqrt(2065)/413). Everything seems correct. Why did I think it was wrong? Let me re-read carefully. u = <-3, 5, 5>, v = <5, 1, -3>. Dot product: -15 + 5 - 15 = -25. Correct. Magnitudes: sqrt(59), sqrt(35). Correct. Cosine: -25/sqrt(2065). Simplified: -5sqrt(2065)/413. Correct. Angle: arccos(...). Correct. The solution is sound.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The magnitude of vector u is calculated incorrectly in line 2. |u| should be sqrt((-3)^2 + 5^2 + 5^2) = sqrt(59), but the dot product calculation in line 1 is also incorrect (-15 -15 + 5 = -25 is correct, wait. Let's recheck. u=<-3,5,5>, v=<5,1,-3>. u.v = -15 + 5 - 15 = -25. Correct. |u| = sqrt(9+25+25) = sqrt(59). Correct. |v| = sqrt(25+1+9) = sqrt(35). Correct. cos(theta) = -25 / (sqrt(59)*sqrt(35)) = -25 / sqrt(2065). The solution simplifies this to -5*sqrt(2065)/413. Let's check: -25/sqrt(2065) * sqrt(2065)/sqrt(2065) = -25*sqrt(2065)/2065. 2065/5 = 413. So -5*sqrt(2065)/413. This is correct. Why did I think it was wrong? Ah, I misread the prompt's provided solution steps. Line 1 says -25. Line 2 says sqrt(59). Line 3 says sqrt(35). Line 4 says -5*sqrt(2065)/413. This is all algebraically correct. Wait, is there a subtle error? The problem asks for the angle. The answer is acos(...). The solution is correct. Let me re-read carefully. u=<-3, 5, 5>. v=<5, 1, -3>. Dot product: -15 + 5 - 15 = -25. |u|^2 = 9+25+25=59. |v|^2 = 25+1+9=35. cos theta = -25/sqrt(2065). Rationalized: -25*sqrt(2065)/2065 = -5*sqrt(2065)/413. Everything seems correct. Is there a 'style' issue? No. Is there a 'misleading' issue? No. It passes.
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 2026-10-13 with SymPy 1.14.0.