Home›Calculus 3›Planes: distances and angles›Problem 9.626

Planes: distances and angles

Problem 9.626 · easy

Find an equation of the plane through \( \displaystyle (3, 3, 3) \) with normal vector \( \displaystyle \left\langle 4, 2, 2 \right\rangle \).

n · (r − P) = 0, i.e. ax + by + cz = n · P.
\[ 2 \cdot 3 + 2 \cdot 3 + 4 \cdot 3 = 24 \]
n · P.✓ Proved

Answer \( 4 x + 2 y + 2 z = 24 \)

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 solution incorrectly states the plane equation as ax + by + cz = n · P, omitting the necessary subtraction of the point's coordinates (ax + by + cz = d where d = n · P). While the final numerical answer is correct, the stated formula is mathematically false and would lead to errors for any point not at the origin.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly states the plane equation as ax + by + cz = n · P, omitting the necessary subtraction of the point's coordinates (ax + by + cz = d where d = n · P). While the final numerical answer is correct, the stated formula is mathematically false and would lead to errors for any point not at the origin.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The setup in line 1 incorrectly maps the normal vector components to the variables (using 'ax + by + cz' implies a=4, b=2, c=2 but the equation in line 2 calculates 2*3 + 2*3 + 4*3, swapping the coefficients for x and z). The correct equation should be 4x + 2y + 2z = 24, but the derivation shown computes 2x + 2y + 4z = 24, which is a different plane.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the point-normal form of a plane equation. The setup is valid, the dot product calculation is correct, and the final equation matches the stated answer.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.