Home›Calculus 3›Planes: distances and angles›Problem 9.717

Planes: distances and angles

Problem 9.717 · easy

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

n · (r − P) = 0, i.e. ax + by + cz = n · P.
\[ 1 \left(-3\right) - 1 \cdot 2 - 1 \left(-1\right) = -4 \]
n · P.✓ Proved

Answer \( - x + y - z = -4 \)

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 calculation of n · P in line 2 is incorrect; it computes -1(-3) + 1(2) - 1(-1) = 6, but the line claims the result is -4. The correct equation should be -x + y - z = 6.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The calculation of n · P in line 2 is incorrect; it computes -1(-3) + 1(2) - 1(-1) = 6, but the line claims the result is -4. The correct equation should be -x + y - z = 6.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution uses the normal vector <1, -1, 1> in the calculation (line 2), which is the negative of the given normal vector <-1, 1, -1>. While the resulting plane equation is equivalent, the setup in line 1 implies using the given normal directly, but the calculation contradicts this by using -n. This is a logical inconsistency in the derivation steps.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The calculation of n · P in line 2 is incorrect; it computes -1*(-3) + 1*(-3) + (-1)*2 = -2, not -4. The correct equation is -x + y - z = -2.
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 2026-10-18 with SymPy 1.14.0.