Home›Calculus 3›Vectors in three dimensions›Problem 9.282

Vectors in three dimensions

Problem 9.282 · easy

For \( \displaystyle \mathbf a = \left\langle 0, 1, 4 \right\rangle \) and \( \displaystyle \mathbf b = \left\langle -3, 4, -6 \right\rangle \), find \( \displaystyle 1\mathbf a - (-4)\mathbf b \) and \( \displaystyle \|\mathbf a - \mathbf b\| \).

\[ \left[\begin{matrix}-12\\17\\-20\end{matrix}\right] \]
Componentwise.✓ Proved
\[ \sqrt{118} \]
‖a − b‖.✓ Proved

Answer \( \left\langle -12, 17, -20 \right\rangle,\ \sqrt{118} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution provides no derivation or calculation steps, only tautological equations that verify the final answer against itself. It fails to demonstrate how the vector arithmetic and norm were computed.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution provides no derivation or calculation steps, only tautological equations that verify the final answer against itself. It fails to demonstrate how the vector arithmetic and norm were computed.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution provides no derivation or calculation steps, merely restating the final answers as tautological equations. It fails to show how the vector arithmetic and norm were computed.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.