Triple products and volumes
Problem 9.330 · easy
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle 0, 4, 0 \right\rangle \), \( \displaystyle \left\langle 1, 3, 1 \right\rangle \) and \( \displaystyle \left\langle -2, -14, -2 \right\rangle \).
- \[ \left[\begin{matrix}8\\0\\-8\end{matrix}\right] \]v × w.✓ Proved
- \[ 0 \]u·(v × w).✓ Proved
- The volume is the absolute value of the triple product — 0 here, so the vectors are coplanar.
Answer \( 0 \)
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the 3 × 3 determinant, computed by mpmath |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The cross product in step 1 is incorrect; the cross product of <1, 3, 1> and <-2, -14, -2> is <8, 0, -8>, but the vectors given in the problem are <0, 4, 0>, <1, 3, 1>, and <-2, -14, -2>. The solution implicitly assumes the first vector is <1, 0, -1> or similar to get a non-zero cross product component, but actually, let's re-calculate. v=<1,3,1>, w=<-2,-14,-2>. v x w = <3(-2)-1(-14), 1(-2)-1(-2), 1(-14)-3(-2)> = <-6+14, -2+2, -14+6> = <8, 0, -8>. This part is correct. Then u=<0,4,0>. u . (v x w) = 0*8 + 4*0 + 0*(-8) = 0. The calculation is actually correct. Wait, let me re-read the vectors. u=<0,4,0>, v=<1,3,1>, w=<-2,-14,-2>. v x w = <8, 0, -8>. u . (v x w) = 0. The volume is 0. The solution is correct. Why did I think it was wrong? Let me check the cross product again. i(3*-2 - 1*-14) - j(1*-2 - 1*-2) + k(1*-14 - 3*-2) = i(-6+14) - j(-2+2) + k(-14+6) = 8i - 0j - 8k. Correct. Dot product with <0,4,0> is 0. Correct. The solution is sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The cross product in step 1 is incorrect; the cross product of <1, 3, 1> and <-2, -14, -2> is <8, 0, -8>, but the vectors given in the problem are <0, 4, 0>, <1, 3, 1>, and <-2, -14, -2>. The solution implicitly assumes the first vector is <1, 0, -1> or similar to get a non-zero cross product component, but actually, let's re-calculate. v=<1,3,1>, w=<-2,-14,-2>. v x w = <3(-2)-1(-14), 1(-2)-1(-2), 1(-14)-3(-2)> = <-6+14, -2+2, -14+6> = <8, 0, -8>. This part is correct. Then u=<0,4,0>. u . (v x w) = 0*8 + 4*0 + 0*(-8) = 0. The calculation is actually correct. Wait, let me re-read the vectors. u=<0,4,0>, v=<1,3,1>, w=<-2,-14,-2>. v x w = <8, 0, -8>. u . (v x w) = 0. The volume is 0. The solution is correct. Why did I think it was wrong? Let me check the cross product again. i(3*-2 - 1*-14) - j(1*-2 - 1*-2) + k(1*-14 - 3*-2) = i(-6+14) - j(-2+2) + k(-14+6) = 8i - 0j - 8k. Correct. Dot product with <0,4,0> is 0. Correct. The solution is sound.qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-07
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/triple_product, checked 2026-10-07 with SymPy 1.14.0.