Cross product
Problem 9.100 · easy
Find \( \displaystyle \mathbf{u} \times \mathbf{v} \) for \( \displaystyle \mathbf{u} = \langle 2, -3, -2 \rangle,\ \mathbf{v} = \langle 2, -1, 2 \rangle \), and the area of the parallelogram they span.
- Expand the determinant with rows i j k, u, v.
- \[ \left[\begin{matrix}-8\\-8\\4\end{matrix}\right] \]The components.✓ Proved
- \[ 12 \]Area = |u × v|.✓ Proved
Answer \( \mathbf{u}\times\mathbf{v} = \langle -8, -8, 4 \rangle,\quad \text{area} = 12 \)
Lines: 2 proved, 1 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | Lagrange's identity √(|u|²|v|² − (u·v)²) gives the same area |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution claims the area is 12, but the magnitude of the cross product vector <-8, -8, 4> is sqrt(64 + 64 + 16) = sqrt(144) = 12. Wait, sqrt(144) is indeed 12. Let me re-read the prompt constraints. 'Find u x v ... and the area'. The stated answer is 12. The solution calculates the cross product components correctly. Then it calculates the magnitude. sqrt((-8)^2 + (-8)^2 + 4^2) = sqrt(64+64+16) = sqrt(144) = 12. The math is correct. Why would it fail? Ah, the problem asks for TWO things: the cross product vector AND the area. The stated answer only provides the area (12). The solution output shows the vector components in line 2, but the 'Stated answer' field in the problem description usually refers to the final answer expected by an automated grader. If the grader expects the vector, 12 is wrong. If the grader expects the area, 12 is right. However, looking at the solution structure: Line 2 outputs the vector. Line 3 outputs 12. The 'Stated answer' in the prompt is '12'. This implies the question might be interpreted as just asking for the area, or the user provided a partial answer. But typically 'Find A and B' requires both. If the 'Stated answer' is the ground truth provided by the system, and it is 12, then the system only cares about the area. But the solution text says 'Find u x v ... and the area'. A complete solution should state the vector. The solution *does* state the vector in line 2. But does it explicitly state 'The cross product is ...'? It just shows the matrix result. More importantly, is the calculation correct? Yes. Is the setup correct? Yes. Is the conclusion correct? Yes. Why would I mark it fail? Let's look closer. Line 2: `Matrix([...])`. This is a representation of the vector. Line 3: `12, 12`. This looks like it's asserting the area is 12. The prompt asks to review the *solution*. The solution provides the vector and the area. The 'Stated answer' is 12. This likely means the 'correct answer' key is 12. If the question asks for two things, and the key is one number, it's ambiguous. However, often in these datasets, if the question asks for multiple things, the answer key might be a tuple or the question is malformed. Let's look for a subtle error. u = <2, -3, -2>, v = <2, -1, 2>. i( (-3)(2) - (-2)(-1) ) = i(-6 - 2) = -8i. j( (2)(2) - (-2)(2) ) = j(4 - (-4)) = 8j. Wait. The j component formula is -( (2)(2) - (-2)(2) ) = -(4+4) = -8. Or using the standard determinant expansion: i(u2v3 - u3v2) - j(u1v3 - u3v1) + k(u1v2 - u2v1). i: (-3)(2) - (-2)(-1) = -6 - 2 = -8. j: -( (2)(2) - (-2)(2) ) = -(4 - (-4)) = -(8) = -8. k: (2)(-1) - (-3)(2) = -2 - (-6) = 4. Vector is <-8, -8, 4>. Magnitude sqrt(64+64+16) = 12. The math is perfect. Is there a style issue? The solution is very brief. 'Expand the determinant...' is a sentence. The equations are marked proved. The logic holds. Perhaps the 'Stated answer' being just 12 implies the user only submitted the area, but the problem asked for the vector too? If so, the solution is incomplete. But the solution *contains* the vector. The 'Stated answer' field in the prompt is likely the 'ground truth' for the final answer. If the ground truth is 12, and the question asks for two things, the ground truth is likely wrong or the question is implicitly asking for the area (common in some contexts where 'area of parallelogram' is the main goal). However, strictly speaking, 'Find u x v ... and the area' requires two outputs. If the solution only highlights 12 as the final result (line 3), it might be considered incomplete. But line 2 is also an equation. Let's look at the 'Stated answer' again. It says 'Stated answer: 12'. This is likely the answer key. If the answer key is 12, then the system expects 12. The solution produces 12. It also produces the vector. This is not an error. It is a 'pass'. Wait, let me check the j-component calculation again carefully. Standard formula: u x v = < u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1 >. u2v3 - u3v2 = (-3)(2) - (-2)(-1) = -6 - 2 = -8. u3v1 - u1v3 = (-2)(2) - (2)(2) = -4 - 4 = -8. u1v2 - u2v1 = (2)(-1) - (-3)(2) = -2 + 6 = 4. Vector <-8, -8, 4>. Correct. Magnitude 12. Correct. Why would this fail? Maybe the 'Stated answer' is just the area, and the solution is correct. I will mark it pass.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution claims the area is 12, but the magnitude of the cross product vector <-8, -8, 4> is sqrt(64 + 64 + 16) = sqrt(144) = 12. Wait, sqrt(144) is indeed 12. Let me re-read the prompt constraints. 'Find u x v ... and the area'. The stated answer is 12. The solution calculates the cross product components correctly. Then it calculates the magnitude. sqrt((-8)^2 + (-8)^2 + 4^2) = sqrt(64+64+16) = sqrt(144) = 12. The math is correct. Why would it fail? Ah, the problem asks for TWO things: the cross product vector AND the area. The stated answer only provides the area (12). The solution output shows the vector components in line 2, but the 'Stated answer' field in the problem description usually refers to the final answer expected by an automated grader. If the grader expects the vector, 12 is wrong. If the grader expects the area, 12 is right. However, looking at the solution structure: Line 2 outputs the vector. Line 3 outputs 12. The 'Stated answer' in the prompt is '12'. This implies the question might be interpreted as just asking for the area, or the user provided a partial answer. But typically 'Find A and B' requires both. If the 'Stated answer' is the ground truth provided by the system, and it is 12, then the system only cares about the area. But the solution text says 'Find u x v ... and the area'. A complete solution should state the vector. The solution *does* state the vector in line 2. But does it explicitly state 'The cross product is ...'? It just shows the matrix result. More importantly, is the calculation correct? Yes. Is the setup correct? Yes. Is the conclusion correct? Yes. Why would I mark it fail? Let's look closer. Line 2: `Matrix([...])`. This is a representation of the vector. Line 3: `12, 12`. This looks like it's asserting the area is 12. The prompt asks to review the *solution*. The solution provides the vector and the area. The 'Stated answer' is 12. This likely means the 'correct answer' key is 12. If the question asks for two things, and the key is one number, it's ambiguous. However, often in these datasets, if the question asks for multiple things, the answer key might be a tuple or the question is malformed. Let's look for a subtle error. u = <2, -3, -2>, v = <2, -1, 2>. i( (-3)(2) - (-2)(-1) ) = i(-6 - 2) = -8i. j( (2)(2) - (-2)(2) ) = j(4 - (-4)) = 8j. Wait. The j component formula is -( (2)(2) - (-2)(2) ) = -(4+4) = -8. Or using the standard determinant expansion: i(u2v3 - u3v2) - j(u1v3 - u3v1) + k(u1v2 - u2v1). i: (-3)(2) - (-2)(-1) = -6 - 2 = -8. j: -( (2)(2) - (-2)(2) ) = -(4 - (-4)) = -(8) = -8. k: (2)(-1) - (-3)(2) = -2 - (-6) = 4. Vector is <-8, -8, 4>. Magnitude sqrt(64+64+16) = 12. The math is perfect. Is there a style issue? The solution is very brief. 'Expand the determinant...' is a sentence. The equations are marked proved. The logic holds. Perhaps the 'Stated answer' being just 12 implies the user only submitted the area, but the problem asked for the vector too? If so, the solution is incomplete. But the solution *contains* the vector. The 'Stated answer' field in the prompt is likely the 'ground truth' for the final answer. If the ground truth is 12, and the question asks for two things, the ground truth is likely wrong or the question is implicitly asking for the area (common in some contexts where 'area of parallelogram' is the main goal). However, strictly speaking, 'Find u x v ... and the area' requires two outputs. If the solution only highlights 12 as the final result (line 3), it might be considered incomplete. But line 2 is also an equation. Let's look at the 'Stated answer' again. It says 'Stated answer: 12'. This is likely the answer key. If the answer key is 12, then the system expects 12. The solution produces 12. It also produces the vector. This is not an error. It is a 'pass'. Wait, let me check the j-component calculation again carefully. Standard formula: u x v = < u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1 >. u2v3 - u3v2 = (-3)(2) - (-2)(-1) = -6 - 2 = -8. u3v1 - u1v3 = (-2)(2) - (2)(2) = -4 - 4 = -8. u1v2 - u2v1 = (2)(-1) - (-3)(2) = -2 + 6 = 4. Vector <-8, -8, 4>. Correct. Magnitude 12. Correct. Why would this fail? Maybe the 'Stated answer' is just the area, and the solution is correct. I will mark it pass.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer returned a non-objectgpt-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/cross_product_area, checked 2026-09-26 with SymPy 1.14.0.