Projections and orthogonality
Problem 9.268 · medium
Find the scalar projection \( \displaystyle \operatorname{comp}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 3, 2, 0 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -3, -2, -2 \right\rangle \).
- \[ -9 - 4 + 0 = -13 \]u·v.✓ Proved
- \[ 4 + 4 + 9 = 17 \]‖v‖².✓ Proved
- \[ - \frac{13 \sqrt{17}}{17} \]comp = u·v / ‖v‖.✓ Proved
Answer \( - \frac{13 \sqrt{17}}{17} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: pass — The solution correctly applies the formula for scalar projection, comp_v u = (u · v) / ||v||, and the arithmetic is verified by the algebra system.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the formula for scalar projection, comp_v u = (u · v) / ||v||, and the arithmetic is verified by the algebra system.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly calculates the dot product and the magnitude of v, and applies the scalar projection formula accurately.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/vector_projection, checked 2026-10-05 with SymPy 1.14.0.