Home›Calculus 3›Projections and orthogonality›Problem 9.430

Projections and orthogonality

Problem 9.430 · easy

Which pairs of \( \displaystyle \mathbf a = \left\langle -4, 3, 2 \right\rangle \), \( \displaystyle \mathbf b = \left\langle -26, -22, -19 \right\rangle \), \( \displaystyle \mathbf c = \left\langle 6, -2, -3 \right\rangle \) are orthogonal?

\[ \left[\begin{matrix}0\\-36\\-55\end{matrix}\right] \]
a·b, a·c, b·c.✓ Proved
Two nonzero vectors are orthogonal exactly when their dot product is 0.

Answer \( \mathbf a \perp \mathbf b \)

Lines: 1 proved, 1 not checked. 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	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	dot products recomputed

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution calculates the dot products but fails to identify which pair is orthogonal. The first entry (a·b) is 0, implying a and b are orthogonal, but the text does not explicitly state this conclusion or check the other pairs (a·c and b·c are non-zero).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-08
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution calculates the dot products but fails to identify which pair is orthogonal. The first entry (a·b) is 0, implying a and b are orthogonal, but the text does not explicitly state this conclusion or check the other pairs (a·c and b·c are non-zero).
qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution computes the dot products but fails to identify which ones are zero. The vector a is orthogonal to c (dot product 0), not b (dot product -36). The stated answer is incorrect.
gpt-oss:20b: fail (style) 2026-10-08 — The solution states the orthogonality criterion correctly, but it does not actually compute or show that a·b=0, nor does it explicitly state the conclusion that a⊥b. The missing evidence and conclusion make the explanation incomplete and unclear for a student.

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 2026-10-08 with SymPy 1.14.0.