Home›Calculus 3›Dot product and angles›Problem 9.141

Dot product and angles

Problem 9.141 · easy

Let \( \displaystyle \mathbf{u} = \langle 2, -2, -1 \rangle \) and \( \displaystyle \mathbf{v} = \langle 3, 2, 0 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-2\right) 2 - 1 \cdot 0 + 2 \cdot 3 = 2 \]
Multiply componentwise and add.✓ Proved
\[ 3 \]
|u|.
\[ \sqrt{13} \]
|v|.
\[ \frac{2 \sqrt{13}}{39} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = 2,\quad \theta = \operatorname{acos}{\left(\frac{2 \sqrt{13}}{39} \right)} \approx 79.34^\circ \)

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 magnitude of vector u is calculated as 3, but ||u|| = sqrt(2^2 + (-2)^2 + (-1)^2) = sqrt(9) = 3 is correct. However, the magnitude of v is sqrt(13). The product of magnitudes is 3*sqrt(13). The cosine is 2/(3*sqrt(13)). The solution claims cos(theta) = 2*sqrt(13)/39, which simplifies to 2/(3*sqrt(13)) only if rationalized incorrectly or if there is a typo in the stated answer vs the derivation. Wait, 2/(3*sqrt(13)) = 2*sqrt(13)/(3*13) = 2*sqrt(13)/39. So the algebra is correct. Let me re-read carefully. Step 2 says |u| = 3. Correct. Step 3 says |v| = sqrt(13). Correct. Step 4 says cos theta = 2*sqrt(13)/39. This is 2 / (3*sqrt(13)) rationalized. Correct. Step 5 says theta = arccos of that. Correct. Wait, is there an error? Problem: u = <2, -2, -1>, v = <3, 2, 0>. u dot v = 6 - 4 + 0 = 2. Correct. |u| = sqrt(4+4+1) = 3. Correct. |v| = sqrt(9+4+0) = sqrt(13). Correct. cos theta = 2 / (3*sqrt(13)). Rationalized: 2*sqrt(13) / 39. Correct. Why would this fail? Let's check the angle calculation. acos(2*sqrt(13)/39) in degrees. 2*sqrt(13)/39 approx 2*3.605/39 approx 7.21/39 approx 0.1848. acos(0.1848) approx 1.384 radians. 1.384 * 180/pi approx 79.3 degrees. The solution seems perfectly correct. Is there a subtle issue? "Multiply componentwise and add" is a bit loose for dot product definition (sum of products of corresponding components), but not wrong. Let's look at the JSON output format required. If it is sound, pass. Is there any ambiguity? "theta = arccos of that" refers to the previous line. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The magnitude of vector u is calculated as 3, but ||u|| = sqrt(2^2 + (-2)^2 + (-1)^2) = sqrt(9) = 3 is correct. However, the magnitude of v is sqrt(13). The product of magnitudes is 3*sqrt(13). The cosine is 2/(3*sqrt(13)). The solution claims cos(theta) = 2*sqrt(13)/39, which simplifies to 2/(3*sqrt(13)) only if rationalized incorrectly or if there is a typo in the stated answer vs the derivation. Wait, 2/(3*sqrt(13)) = 2*sqrt(13)/(3*13) = 2*sqrt(13)/39. So the algebra is correct. Let me re-read carefully. Step 2 says |u| = 3. Correct. Step 3 says |v| = sqrt(13). Correct. Step 4 says cos theta = 2*sqrt(13)/39. This is 2 / (3*sqrt(13)) rationalized. Correct. Step 5 says theta = arccos of that. Correct. Wait, is there an error? Problem: u = <2, -2, -1>, v = <3, 2, 0>. u dot v = 6 - 4 + 0 = 2. Correct. |u| = sqrt(4+4+1) = 3. Correct. |v| = sqrt(9+4+0) = sqrt(13). Correct. cos theta = 2 / (3*sqrt(13)). Rationalized: 2*sqrt(13) / 39. Correct. Why would this fail? Let's check the angle calculation. acos(2*sqrt(13)/39) in degrees. 2*sqrt(13)/39 approx 2*3.605/39 approx 7.21/39 approx 0.1848. acos(0.1848) approx 1.384 radians. 1.384 * 180/pi approx 79.3 degrees. The solution seems perfectly correct. Is there a subtle issue? "Multiply componentwise and add" is a bit loose for dot product definition (sum of products of corresponding components), but not wrong. Let's look at the JSON output format required. If it is sound, pass. Is there any ambiguity? "theta = arccos of that" refers to the previous line. I will mark it as pass.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The calculation of |u| is incorrect; the solution claims |u|=3, but |u|=sqrt(2^2+(-2)^2+(-1)^2)=3 is actually correct. However, the calculation of |
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The magnitude of vector u is calculated as 3, but it should be sqrt(9) = 3. Wait, u = <2, -2, -1>, so |u|^2 = 4+4+1=9, |u|=3. That is correct. The magnitude of v = <3, 2, 0> is sqrt(9+4+0)=sqrt(13). That is correct. The dot product is 2*3 + (-2)*2 + (-1)*0 = 6-4+0=2. That is correct. cos(theta) = 2 / (3*sqrt(13)) = 2*sqrt(13)/39. That is correct. The final answer is arccos(2*sqrt(13)/39). The solution steps are algebraically correct. Why did I think there was an error? Let me re-read carefully. Step 1: -2*2 - 1*0 + 2*3. This corresponds to u_y*v_y + u_z*v_z + u_x*v_x? No, standard order is u_x*v_x + u_y*v_y + u_z*v_z. u_x=2, v_x=3 -> 6. u_y=-2, v_y=2 -> -4. u_z=-1, v_z=0 -> 0. Sum = 2. The expression written is -2*2 - 1*0 + 2*3. This is u_y*v_y + u_z*v_z + u_x*v_x. The order is mixed up but the terms are correct components. It evaluates to 2. Step 2: |u| = 3. Correct. Step 3: |v| = sqrt(13). Correct. Step 4: cos theta = 2 / (3*sqrt(13)). Rationalized: 2*sqrt(13)/39. Correct. Step 5: theta = arccos(...). Correct. The solution is actually correct. I will mark it as pass.
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.