Planes: distances and angles
Problem 9.719 · easy
Find the distance from \( \displaystyle (3, 4, 0) \) to the plane \( \displaystyle - 2 x + 3 y - 2 z = 4 \).
- Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.Reviewed
- \[ \frac{\left|{\left(-2\right) 3 - 4 - 2 \cdot 0 + 3 \cdot 4}\right|}{\sqrt{\left(-2\right)^{2} + \left(-2\right)^{2} + 3^{2}}} = \frac{2 \sqrt{17}}{17} \]Substitute.✓ Proved
Answer \( \frac{2 \sqrt{17}}{17} \)
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the distance formula and substitutes the values accurately. The algebraic simplification in the equation line is correct, leading to the right final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the distance formula and substitutes the values accurately. The algebraic simplification in the equation line is correct, leading to the right final answer.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The formula in step 1 is incorrect; it should be |ax₀ + by₀ + cz₀ + d| / ‖n‖ for a plane ax + by + cz = -d, or |ax₀ + by₀ + cz₀ - d| / ‖n‖ for ax + by + cz = d. The equation given is -2x + 3y - 2z = 4, so d=4. The numerator should be |-2(3) + 3(4) - 2(0) - 4| = |-6 + 12 - 0 - 4| = |2| = 2. The solution calculates |-4 + (-2*3 - 2*0 + 3*4)| which is |-4 + 6| = 2, coincidentally getting the right magnitude but using a confusing and formally incorrect expression for the numerator that mixes signs arbitrarily. Specifically, the standard formula is |Ax_0 + By_0 + Cz_0 + D| / sqrt(A^2+B^2+C^2) for Ax+By+Cz+D=0. Here -2x+3y-2z-4=0, so numerator is |-6+12-0-4|=2. The solution's text says 'ax0+by0+cz0-d' implying the plane is ax+by+cz=d, which is correct, but the substitution line is messy and potentially misleading about the sign of d. However, the bigger issue is that the denominator calculation in the equation line uses sqrt((-2)**2 + (-2)**2 + 3**2). The normal vector is <-2, 3, -2>. The magnitude squared is 4 + 9 + 4 = 17. The solution calculates sqrt(4 + 4 + 9) = sqrt(17). Wait, the coefficients are -2, 3, -2. The solution writes (-2)**2 + (-2)**2 + 3**2. This squares -2 twice and 3 once. The z-coefficient is -2, so (-2)^2 is 4. The y-coefficient is 3, so 3^2 is 9. The x-coefficient is -2, so (-2)^2 is 4. The sum is 4+9+4=17. The solution writes (-2)**2 + (-2)**2 + 3**2. This corresponds to x^2 + z^2 + y^2? Or x^2 + y^2 + z^2? If it's x=-2, y=3, z=-2, then terms are 4, 9, 4. The solution has 4, 4, 9. This is just a permutation of the terms, so the value is correct. But let's look closer at the numerator. The solution writes Abs(-4 + (-2*3 - 2*0 + 3*4)). This is Abs(-4 + (-6 - 0 + 12)) = Abs(-4 + 6) = 2. This is correct. Is there an error? Let's re-read the sentence. 'Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖'. This assumes the plane is ax+by+cz=d. The plane is -2x+3y-2z=4. So a=-2, b=3, c=-2, d=4. Numerator: |-2(3) + 3(4) - 2(0) - 4| = |-6 + 12 - 4| = 2. Denominator: sqrt(4+9+4) = sqrt(17). Result 2/sqrt(17) = 2sqrt(17)/17. The solution is actually correct. Why did I think it was wrong? Let me check the denominator again. The solution writes sqrt((-2)**2 + (-2)**2 + 3**2). The coefficients are a=-2, b=3, c=-2. The squares are 4, 9, 4. The solution sums 4 + 4 + 9. This is correct. The numerator is correct. The result is correct. Is there a subtle error? The sentence says 'Distance = ...'. It is a standard formula. The substitution is correct. The answer is correct. I will mark it as pass.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The substitution in line 2 is incorrect. The formula requires |ax₀ + by₀ + cz₀ - d|, but the numerator calculates |-4 + (-6 - 0 + 12)| = |2|, whereagpt-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.