Home›Calculus 3›Planes: distances and angles›Problem 9.889

Planes: distances and angles

Problem 9.889 · easy

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

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
Reviewed
\[ \frac{\left|{\left(-3\right) 4 - 2 \cdot 2 + 4 - 4 \left(-3\right)}\right|}{\sqrt{\left(-2\right)^{2} + \left(-3\right)^{2} + \left(-4\right)^{2}}} = 0 \]
Substitute.✓ Proved

Answer \( 0 \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and plane coefficients is accurate, and the arithmetic correctly yields a distance of 0.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and plane coefficients is accurate, and the arithmetic correctly yields a distance of 0.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly applies the point-to-plane distance formula. The substitution and arithmetic are correct, yielding a distance of 0, which is consistent with the point lying on the plane.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: fail (error) 2026-11-04 — The formula in step 1 uses 'd' as the constant term on the RHS of the plane equation (ax+by+cz=d), but the substitution in step 2 incorrectly adds 4 instead of subtracting it (or vice versa depending on sign convention), leading to a numerator of 0. The correct distance is non-zero. Specifically, for -2x-3y-4z=-4, the distance is |(-2)(2) + (-3)(4) + (-4)(-3) - (-4)| / sqrt(29) = |-4 -12 + 12 + 4| / sqrt(29) = 0. Wait, let me re-calculate. Point (2, 4, -3). Plane -2x -3y -4z = -4. LHS = -2(2) -3(4) -4(-3) = -4 -12 + 12 = -4. RHS = -4. The point IS on the plane. So the distance IS 0. The calculation in step 2 is: Abs((-3*4 - 2*2 - 4*(-3)) + 4). Let's check the terms. -3*4 = -12. -2*2 = -4. -4*(-3) = 12. Sum = -12 -4 + 12 = -4. Then +4 = 0. The calculation is correct. The formula is standard. The result is correct. Why did I think it was wrong? Let me re-read carefully. Step 1: Distance = |ax0 + by0 + cz0 - d| / ||n||. Step 2 substitutes. The plane is -2x -3y -4z = -4. So a=-2, b=-3, c=-4, d=-4. The formula uses d. So we calculate |(-2)(2) + (-3)(4) + (-4)(-3) - (-4)|. This is |-4 -12 + 12 + 4| = 0. The solution writes: Abs((-3*4 - 2*2 - 4*(-3)) + 4). This corresponds to |(-3)(4) + (-2)(2) + (-4)(-3) - (-4)|? No, it has +4. If d=-4, then -d = +4. So the term is +4. The calculation is correct. The verdict should be pass.
gpt-oss:20b: pass 2026-11-04

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 2026-11-05 with SymPy 1.14.0.