Home›Calculus 3›Projections and orthogonality›Problem 9.347

Projections and orthogonality

Problem 9.347 · easy

Which pairs of \( \displaystyle \mathbf a = \left\langle -4, -4, 2 \right\rangle \), \( \displaystyle \mathbf b = \left\langle 20, -34, -28 \right\rangle \), \( \displaystyle \mathbf c = \left\langle -4, -4, -4 \right\rangle \) are orthogonal?

\[ \left[\begin{matrix}0\\24\\168\end{matrix}\right] \]
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 explicitly state which pairs are orthogonal based on the results. Specifically, it does not identify that a·c = 0, meaning a and c are orthogonal, while a·b ≠ 0 and b·c ≠ 0. The stated answer only lists a and b, which is incorrect as their dot product is not zero.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution calculates the dot products but fails to explicitly state which pairs are orthogonal based on the results. Specifically, it does not identify that a·c = 0, meaning a and c are orthogonal, while a·b ≠ 0 and b·c ≠ 0. The stated answer only lists a and b, which is incorrect as their dot product is not zero.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: inconclusive 2026-10-07 — reviewer returned a non-object

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.