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Planes: distances and angles

Problem 9.775 · easy

Find an equation of the plane through \( \displaystyle (3, 2, -3) \) with normal vector \( \displaystyle \left\langle 1, 1, -3 \right\rangle \).

n · (r − P) = 0, i.e. ax + by + cz = n · P.
\[ 1 \cdot 2 + 1 \cdot 3 - 3 \left(-3\right) = 14 \]
n · P.✓ Proved

Answer \( x + y - 3 z = 14 \)

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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ 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 plane holds P and its gradient is parallel to n

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The dot product calculation in line 2 is incorrect; it computes 1*2 + 1*3 - 3*(-3) = 14, but the point is (3, 2, -3), so the correct calculation is 1*3 + 1*2 - 3*(-3) = 14. Although the final numerical result happens to be correct due to the symmetry of the x and y components, the intermediate step explicitly uses the wrong coordinates for the dot product.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The dot product calculation in line 2 is incorrect; it computes 1*2 + 1*3 - 3*(-3) = 14, but the point is (3, 2, -3), so the correct calculation is 1*3 + 1*2 - 3*(-3) = 14. Although the final numerical result happens to be correct due to the symmetry of the x and y components, the intermediate step explicitly uses the wrong coordinates for the dot product.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution swaps the x and y coordinates of the point P=(3, 2, -3) when computing the dot product n · P. It calculates 1*2 + 1*3 instead of 1*3 + 1*2, leading to an incorrect constant term.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The calculation of n · P is incorrect; it uses the coordinates (2, 3, -3) instead of the given point (3, 2, -3). The correct dot product is 1(3) + 1(2) - 3(-3) = 14, so while the final constant happens to be correct by coincidence, the intermediate step shown is mathematically wrong.
gpt-oss:20b: pass 2026-10-15

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/planes_distance_angle, checked 2026-10-19 with SymPy 1.14.0.