Home›Calculus 3›Projections and orthogonality›Problem 9.346

Projections and orthogonality

Problem 9.346 · easy

Which pairs of \( \displaystyle \mathbf a = \left\langle 3, 5, 4 \right\rangle \), \( \displaystyle \mathbf b = \left\langle 34, -18, -3 \right\rangle \), \( \displaystyle \mathbf c = \left\langle 0, -2, 4 \right\rangle \) are orthogonal?

\[ \left[\begin{matrix}0\\6\\24\end{matrix}\right] \]
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 computed dot products show that a·c = 0, meaning vectors a and c are orthogonal. However, the stated answer claims a and b are orthogonal, which contradicts the calculation (a·b = 6 ≠ 0).

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The computed dot products show that a·c = 0, meaning vectors a and c are orthogonal. However, the stated answer claims a and b are orthogonal, which contradicts the calculation (a·b = 6 ≠ 0).
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The stated answer claims a is orthogonal to b, but the computed dot product a·b is 6, not 0. The vectors a and c are orthogonal (a·c = 0), so the conclusion is factually incorrect.

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.