Home›Calculus 3›Projections and orthogonality›Problem 9.265

Projections and orthogonality

Problem 9.265 · medium

Find \( \displaystyle \operatorname{proj}_{\mathbf v} \mathbf u \) for \( \displaystyle \mathbf u = \left\langle 1, -2 \right\rangle \), \( \displaystyle \mathbf v = \left\langle -1, -4 \right\rangle \).

\[ -1 + 8 = 7 \]
u·v.✓ Proved
\[ 1 + 16 = 17 \]
‖v‖².✓ Proved
\[ \left[\begin{matrix}- \frac{7}{17}\\- \frac{28}{17}\end{matrix}\right] \]
proj = (u·v/‖v‖²) v.✓ Proved

Answer \( \left\langle - \frac{7}{17}, - \frac{28}{17} \right\rangle \)

✓ 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	the remainder is orthogonal to v and the projection is parallel to it

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the vector projection formula and the arithmetic is verified.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the vector projection formula and the arithmetic is verified.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the vector projection formula. The dot product and squared magnitude are calculated correctly, and the final vector matches the stated answer.
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.