Projections and orthogonality
Problem 9.349 · medium
Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle -1, 1 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -2, 4 \right\rangle \).
- \[ 2 + 4 = 6 \]u·v.✓ Proved
- \[ 4 + 16 = 20 \]‖v‖².✓ Proved
- \[ \frac{3 \sqrt{5}}{5} \]comp = u·v / ‖v‖.✓ Proved
Answer \( \frac{3 \sqrt{5}}{5} \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the formula for scalar projection or calculate the magnitude of v, jumping directly to the final answer. While the arithmetic is correct, the lack of intermediate steps (specifically calculating ||v|| = sqrt(20) = 2*sqrt(5)) makes the derivation opaque and does not demonstrate the method.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly state the formula for scalar projection or calculate the magnitude of v, jumping directly to the final answer. While the arithmetic is correct, the lack of intermediate steps (specifically calculating ||v|| = sqrt(20) = 2*sqrt(5)) makes the derivation opaque and does not demonstrate the method.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly calculates the dot product and the magnitude of v, leading to the correct scalar projection. Although the intermediate steps are presented as unchecked equations rather than derived sentences, the mathematical logic is sound and the final answer is correct.
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 2026-10-07 with SymPy 1.14.0.