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Vectors in the plane

Problem 9.271 · easy

Find the unit vector in the direction of \( \displaystyle \mathbf v = \left\langle -2, -2 \right\rangle \).

\[ 2 \sqrt{2} \]
‖v‖.✓ Proved
\[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{2}}{2}\\\frac{\left(-1\right) \sqrt{2}}{2}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{2}}{2}\\- \frac{\sqrt{2}}{2}\end{matrix}\right] \]
v/‖v‖.✓ Proved

Answer \( \left\langle - \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2} \right\rangle \)

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	length 1 and parallel to v, same direction

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly calculates the magnitude and normalizes the vector. Although the output format uses a column matrix instead of the angle bracket notation in the problem statement, the mathematical content is correct and unambiguous.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the magnitude and normalizes the vector. Although the output format uses a column matrix instead of the angle bracket notation in the problem statement, the mathematical content is correct and unambiguous.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution provides no explanatory sentences to justify the steps or define the method for finding a unit vector. It consists entirely of unchecked algebraic identities, failing to demonstrate the logical connection between the vector, its magnitude, and the final unit vector.
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_plane, checked 2026-10-05 with SymPy 1.14.0.