Home›Calculus 3›Planes: distances and angles›Problem 9.640

Planes: distances and angles

Problem 9.640 · easy

Find the distance from \( \displaystyle (2, -4, 1) \) to the plane \( \displaystyle x + 4 y + 3 z = 2 \).

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
\[ \frac{\left|{4 \left(-4\right) - 2 + 1 \cdot 2 + 3 \cdot 1}\right|}{\sqrt{1^{2} + 3^{2} + 4^{2}}} = \frac{\sqrt{26}}{2} \]
Substitute.✓ Proved

Answer \( \frac{\sqrt{26}}{2} \)

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 foot of the perpendicular found numerically, then the distance to it

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The substitution in step 2 incorrectly assigns coefficients to the coordinates (using 4 for x, 1 for y, 3 for z instead of 1, 4, 3). Although the final numerical answer is coincidentally correct due to the symmetry of the squared terms in the denominator and the specific values, the method demonstrated is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The substitution in step 2 incorrectly assigns coefficients to the coordinates (using 4 for x, 1 for y, 3 for z instead of 1, 4, 3). Although the final numerical answer is coincidentally correct due to the symmetry of the squared terms in the denominator and the specific values, the method demonstrated is wrong.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The substitution in line 2 incorrectly assigns the coefficients a, b, c to the wrong variables (using 4 for x, 1 for y, 3 for z instead of 1, 4, 3). Although the final numerical answer is coincidentally correct because the set of values {1, 3, 4} is the same, the method demonstrated is wrong and would fail for general inputs.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The substitution in step 2 incorrectly assigns the coefficients a, b, c to the wrong components of the normal vector (using 4 for x, 1 for y, 3 for z instead of 1, 4, 3). Although the final numerical answer is coincidentally correct because the sum of squares and the linear combination happen to yield the same magnitude, the setup is fundamentally wrong.
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.