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Projections and orthogonality

Problem 9.505 · medium

Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 2, -4 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -5, -5 \right\rangle \).

\[ -10 + 20 = 10 \]
u·v.✓ Proved
\[ 25 + 25 = 50 \]
‖v‖².✓ Proved
\[ \sqrt{2} \]
comp = u·v / ‖v‖.✓ Proved

Answer \( \sqrt{2} \)

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
3	✓ 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	‖u‖ cos θ with θ from arccos

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly calculates the magnitude of v as sqrt(2) instead of 5*sqrt(2), leading to an incorrect final answer. The correct scalar projection is 10 / (5*sqrt(2)) = sqrt(2).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly calculates the magnitude of v as sqrt(2) instead of 5*sqrt(2), leading to an incorrect final answer. The correct scalar projection is 10 / (5*sqrt(2)) = sqrt(2).
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly calculates the magnitude of v as sqrt(2) instead of 5*sqrt(2), leading to an incorrect final answer.

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/vector_projection, checked 2026-10-11 with SymPy 1.14.0.