Home›Calculus 3›Dot product and angles›Problem 9.10

Dot product and angles

Problem 9.10 · easy

Let \( \displaystyle \mathbf{u} = \langle 0, -1, -5 \rangle \) and \( \displaystyle \mathbf{v} = \langle 2, -2, 1 \rangle \). Find \( \displaystyle \mathbf{u} \cdot \mathbf{v} \) and the angle between them.

\[ \left(-5\right) 1 + 0 \cdot 2 - 1 \left(-2\right) = -3 \]
Multiply componentwise and add.✓ Proved
\[ \sqrt{26} \]
|u|.
\[ 3 \]
|v|.
\[ \frac{\left(-1\right) \sqrt{26}}{26} = - \frac{\sqrt{26}}{26} \]
cos θ = u·v / (|u||v|).
θ = arccos of that.

Answer \( \mathbf{u}\cdot\mathbf{v} = -3,\quad \theta = \operatorname{acos}{\left(- \frac{\sqrt{26}}{26} \right)} \approx 101.31^\circ \)

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 v is calculated as 3, but the correct magnitude is sqrt(9) = 3. Wait, let's recheck. v = <2, -2, 1>. |v|^2 = 4 + 4 + 1 = 9. So |v| = 3. This is correct. Let's recheck u. u = <0, -1, -5>. |u|^2 = 0 + 1 + 25 = 26. So |u| = sqrt(26). This is correct. Dot product: 0*2 + (-1)*(-2) + (-5)*1 = 0 + 2 - 5 = -3. This is correct. cos(theta) = -3 / (3*sqrt(26)) = -1/sqrt(26) = -sqrt(26)/26. This is correct. The angle is arccos(-sqrt(26)/26). The solution seems correct. Why did I think it was wrong? Let me re-read the prompt carefully. 'Find u . v and the angle between them.' The solution calculates the dot product as -3. It calculates the angle. The stated answer is just the angle. The solution steps are correct. Is there a subtle error? Step 1 says '-5*1 + 0*2 - 1*(-2)'. This corresponds to components (-5, 0, -1) and (1, 2, -2)? No, u=<0,-1,-5>, v=<2,-2,1>. The dot product is 0*2 + (-1)*(-2) + (-5)*1. The equation in step 1 is written as '-5*1 + 0*2 - 1*(-2)'. This is -5 + 0 + 2 = -3. The terms are permuted but the values are correct. 0*2 is the first term in standard order, -1*-2 is the second, -5*1 is the third. The equation writes them in a different order: -5*1 (third component product), 0*2 (first component product), -1*-2 (second component product). This is algebraically valid. Step 2: |u| = sqrt(26). Correct. Step 3: |v| = 3. Correct. Step 4: cos theta = -3 / (3*sqrt(26)) = -1/sqrt(26) = -sqrt(26)/26. Correct. Step 5: theta = arccos(...). Correct. The stated answer matches. It seems the solution is actually correct. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The magnitude of vector v is calculated as 3, but the correct magnitude is sqrt(9) = 3. Wait, let's recheck. v = <2, -2, 1>. |v|^2 = 4 + 4 + 1 = 9. So |v| = 3. This is correct. Let's recheck u. u = <0, -1, -5>. |u|^2 = 0 + 1 + 25 = 26. So |u| = sqrt(26). This is correct. Dot product: 0*2 + (-1)*(-2) + (-5)*1 = 0 + 2 - 5 = -3. This is correct. cos(theta) = -3 / (3*sqrt(26)) = -1/sqrt(26) = -sqrt(26)/26. This is correct. The angle is arccos(-sqrt(26)/26). The solution seems correct. Why did I think it was wrong? Let me re-read the prompt carefully. 'Find u . v and the angle between them.' The solution calculates the dot product as -3. It calculates the angle. The stated answer is just the angle. The solution steps are correct. Is there a subtle error? Step 1 says '-5*1 + 0*2 - 1*(-2)'. This corresponds to components (-5, 0, -1) and (1, 2, -2)? No, u=<0,-1,-5>, v=<2,-2,1>. The dot product is 0*2 + (-1)*(-2) + (-5)*1. The equation in step 1 is written as '-5*1 + 0*2 - 1*(-2)'. This is -5 + 0 + 2 = -3. The terms are permuted but the values are correct. 0*2 is the first term in standard order, -1*-2 is the second, -5*1 is the third. The equation writes them in a different order: -5*1 (third component product), 0*2 (first component product), -1*-2 (second component product). This is algebraically valid. Step 2: |u| = sqrt(26). Correct. Step 3: |v| = 3. Correct. Step 4: cos theta = -3 / (3*sqrt(26)) = -1/sqrt(26) = -sqrt(26)/26. Correct. Step 5: theta = arccos(...). Correct. The stated answer matches. It seems the solution is actually correct. I will mark it as pass.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] The magnitude of vector v is calculated as 3, but |v| = sqrt(2^2 + (-2)^2 + 1^2) = sqrt(9) = 3 is correct. However, the dot product calculation in step 1 is -5*1 + 0*2 + (-1)*(-2) = -5 + 0 + 2 = -3. The solution lists the result as -3, which is correct. Wait, let me re-read step 1. It says '-5*1 + 0*2 - 1*(-2)'. The z-component of u is -5, v is 1. The y-component of u is -1, v is -2. The x-component of u is 0, v is 2. So u·v = 0*2 + (-1)*(-2) + (-5)*1 = 0 + 2 - 5 = -3. The equation shows '-5*1 + 0*2 - 1*(-2)' which evaluates to -5 + 0 + 2 = -3. This is correct. Step 2: |u| = sqrt(0^2 + (-1)^2 + (-5)^2) = sqrt(1+25) = sqrt(26). Correct. Step 3: |v| = sqrt(2^2 + (-2)^2 + 1^2) = sqrt(4+4+1) = sqrt(9) = 3. Correct. Step 4: cos(theta) = -3 / (3*sqrt(26)) = -1/sqrt(26) = -sqrt(26)/26. Correct. Step 5: theta = arccos(-sqrt(26)/26). Correct. Why did I think it was wrong? Let me check the prompt's stated answer. acos(-sqrt(26)/26). The solution leads to this. Is there a subtle error? The prompt asks to find u·v AND the angle. The solution calculates u·v in step 1, but the final answer only provides the angle. The 'Stated answer' in the problem description is just the angle. Usually, 'Find A and B' requires both in the final output. However, the review instructions say 'check that the setup models the problem... each conclusion follows'. Step 5 concludes theta. It does not explicitly state the dot product as a final answer, but it is computed. Is this an error? Or just incomplete? The 'Stated answer' provided in the problem block is only the angle. This suggests the problem might only care about the angle, or the stated answer is partial. But looking closely at Step 1: '-5*1 + 0*2 - 1*(-2)'. The term '-1*(-2)' corresponds to the y-components. u_y = -1, v_y = -2. Product is 2. The expression writes '- 1 * (-2)'. This is mathematically equivalent to +2. But usually one writes u_y * v_y. Here it writes -1 * -2. It's fine. Let's look at Step 3 again. |v| = 3. Step 4: -sqrt(26)/26. This is -1/sqrt(26). -3 / (3*sqrt(26)) = -1/sqrt(26). Correct. Is there a domain error? No. Is the angle unique? Yes, [0, pi]. I see no mathematical error. Why would it fail? Maybe the 'Stated answer' is considered the target, and the solution matches it. Let me re-read the 'Stated answer' line. 'Stated answer: acos(-sqrt(26)/26)'. The solution ends with 'theta = arccos of that'. It doesn't explicitly write the final value. But Step 4 has the value. Step 5 says 'arccos of that'. This is a bit informal but correct. Is there a hidden trap? u = <0, -1, -5>, v = <2, -2, 1>. Dot product: 0 + 2 - 5 = -3. |u| = sqrt(26). |v| = 3. cos = -3/(3sqrt(26)) = -1/sqrt(26). Angle = arccos(-1/sqrt(26)). The solution is correct. Wait, look at Step 1 again. '-5*1 + 0*2 - 1*(-2)'. The order of terms is z, x, y. This is fine. The result is -3. The label says '-3'. Correct. Step 2: sqrt(26). Correct. Step 3: 3. Correct. Step 4: -sqrt(26)/26. Correct. Step 5: Sentence. It's a bit vague ('of that'), but logically follows. I will mark it pass. Wait, I must be adversarial. Is 'arccos' the same as 'acos'? Yes. Is the range correct? Yes. Is the simplification correct? Yes. I see no error. I will pass it.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (style) 2026-09-26 — [domain objection, downgraded to style] The magnitude of vector v is calculated as 3, but the correct magnitude is sqrt(9) = 3? No, |v| = sqrt(2^2 + (-2)^2 + 1^2) = sqrt(4+4+1) = sqrt(9) = 3. Wait, let me re-read the solution. Line 3 says '3, 3 -- |v|'. This implies |v|=3. Let's check |u|. |u| = sqrt(0^2 + (-1)^2 + (-5)^2) = sqrt(1+25) = sqrt(26). Line 2 says 'sqrt(26), sqrt(26) -- |u|'. This is correct. Line 1 calculates u.v = 0*2 + (-1)*(-2) + (-5)*1 = 0 + 2 - 5 = -3. The solution says '-3'. This is correct. Line 4 calculates cos(theta) = -3 / (sqrt(26)*3) = -1/sqrt(26) = -sqrt(26)/26. This is correct. Line 5 says theta = arccos of that. The stated answer is acos(-sqrt(26)/26). This matches. Why did I think it was wrong? Let me re-read carefully. Ah, I see. The problem asks for 'the angle between them'. The solution provides the formula for the angle. Is there a subtle error? No, the math seems correct. Wait, let me look at the JSON output format again. I need to find what is WRONG. Is there a style issue? 'acos' vs 'arccos'? No. Is the magnitude of v wrong? v = <2, -2, 1>. 4+4+1=9. sqrt(9)=3. Correct. u = <0, -1, -5>. 0+1+25=26. sqrt(26). Correct. Dot product: 0 + 2 - 5 = -3. Correct. Cos theta = -3 / (3*sqrt(26)) = -1/sqrt(26). Correct. The solution is actually correct. Did I miss something? Let me check the prompt again. 'Find u . v and the angle between them.' The solution finds the dot product (line 1) and the angle (line 5). The stated answer only gives the angle. Usually, if a problem asks for two things, the answer should provide both. However, the 'Stated answer' field in the prompt is just 'acos(-sqrt(26)/26)'. The solution derives this. Is the solution incomplete because it doesn't explicitly state the dot product as a final answer? The prompt asks to review the *solution*. The solution calculates the dot product in line 1. It calculates the angle in line 5. The 'Stated answer' provided in the prompt context might just be the final part. But wait, looking at line 1: '-5*1 + 0*2 - 1*(-2), -3'. The order of multiplication is u_z*v_z + u_x*v_x + u_y*v_y. u=<0,-1,-5>, v=<2,-2,1>. u.x*v.x = 0*2=0. u.y*v.y = -1*-2=2. u.z*v.z = -5*1=-5. Sum = -3. The line writes '-5*1 + 0*2 - 1*(-2)'. This corresponds to u.z*v.z + u.x*v.x + u.y*v.y. This is a valid permutation. The result is -3. Everything seems correct. Why would it fail? Maybe the 'Stated answer' is considered the ground truth for the *final* output, and the solution is just the derivation. If the solution is correct, I should pass it. Let me double check if I am being adversarial enough. Is there a domain issue? arccos range is [0, pi]. The angle between vectors is defined in [0, pi]. This is standard. Is the notation 'acos' acceptable? Yes. Is the simplification -1/sqrt(26) to -sqrt(26)/26 required? It is done. I see no error. Wait, look at line 3 again. '3, 3 -- |v|'. The first '3' is the calculated value, the second is the expected value? Or is it 'result, expected'? The format is [equation, proved] result, expected. So it claims |v|=3. This is correct. I will pass it.
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.