Vectors in three dimensions
Problem 9.367 · easy
For \( \displaystyle \mathbf a = \left\langle 0, -3, 4 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle 3, -4, 6 \right\rangle \), find \( \displaystyle -4\mathbf a - (3)\mathbf b \) and \( \displaystyle \|\mathbf a - \mathbf b\| \).
- \[ \left[\begin{matrix}-9\\24\\-34\end{matrix}\right] \]Componentwise.✓ Proved
- \[ \sqrt{14} \]‖a − b‖.✓ Proved
Answer \( \left\langle -9, 24, -34 \right\rangle,\ \sqrt{14} \)
Every line of this solution was proved by the computer algebra system SymPy. 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the linear combination and the norm, matching the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly computes the linear combination and the norm, matching the stated answer.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution provides no derivation or calculation steps, merely restating the final answer as an identity. It fails to show how the vector arithmetic and norm were computed.gpt-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/vectors_space, checked 2026-10-07 with SymPy 1.14.0.