Home›Calculus 3›Vectors in three dimensions›Problem 9.364

Vectors in three dimensions

Problem 9.364 · easy

For $ \displaystyle \mathbf a = \left\langle 3, -4, 0 \right\rangle $ and $ \displaystyle \mathbf b = \left\langle -2, 3, -6 \right\rangle $, find $ \displaystyle 2\mathbf a - (-1)\mathbf b $ and $ \displaystyle \|\mathbf a - \mathbf b\| $.

\[ \left[\begin{matrix}4\\-5\\-6\end{matrix}\right] \]
Componentwise.✓ Proved
\[ \sqrt{110} \]
‖a − b‖.✓ Proved

Answer $ \left\langle 4, -5, -6 \right\rangle,\ \sqrt{110} $

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 fails to show the calculation for the vector $2\mathbf{a} - (-1)\mathbf{b}$. The first equation checks the final result against itself rather than deriving it from the given vectors $\mathbf{a}$ and $\mathbf{b}$.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to show the calculation for the vector $2\mathbf{a} - (-1)\mathbf{b}$. The first equation checks the final result against itself rather than deriving it from the given vectors $\mathbf{a}$ and $\mathbf{b}$.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to show the calculation for the vector $2\mathbf{a} - (-1)\mathbf{b}$, providing only the final result without derivation. Additionally, the label for the second equation is misleading as it does not show the intermediate step of computing $\mathbf{a} - \mathbf{b}$.
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.