Triple products and volumes
Problem 9.563 · easy
Find the volume of the parallelepiped determined by \( \displaystyle \left\langle -3, -3, 1 \right\rangle \), \( \displaystyle \left\langle 4, 1, -1 \right\rangle \) and \( \displaystyle \left\langle 2, -4, 0 \right\rangle \).
- \[ \left[\begin{matrix}-4\\-2\\-18\end{matrix}\right] \]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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: inconclusive 2026-10-13 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The cross product calculation in step 1 is incorrect. The cross product of <4, 1, -1> and <2, -4, 0> is <-4, -2, -18>, but the scalar triple product with <-3, -3, 1> is (-3)(-4) + (-3)(-2) + (1)(-18) = 12 + 6 - 18 = 0. Wait, the calculation in step 1 is actually correct for v x w. Let me re-read the prompt's provided solution lines. Line 1 says v x w = <-4, -2, -18>. Line 2 says u . (v x w) = 0. Let's check u . (v x w) = <-3, -3, 1> . <-4, -2, -18> = 12 + 6 - 18 = 0. The math is correct. Why did I think it was wrong? Ah, I need to check if the vectors are indeed coplanar. Det = 0. So volume is 0. The solution is correct. Wait, let me double check the cross product. v=<4,1,-1>, w=<2,-4,0>. i(0-4) - j(0-(-2)) + k(-16-2) = -4i - 2j - 18k. Correct. Dot product with u=<-3,-3,1>: 12 + 6 - 18 = 0. Correct. The verdict should be pass.
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 2026-10-13 with SymPy 1.14.0.