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Planes: distances and angles

Problem 9.810 · easy

Find the distance from \( \displaystyle (1, 2, 1) \) to the plane \( \displaystyle 3 x + 3 y + 3 z = -5 \).

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
\[ \frac{\left|{3 \cdot 1 + 3 \cdot 1 + 5 + 3 \cdot 2}\right|}{\sqrt{3^{2} + 3^{2} + 3^{2}}} = \frac{17 \sqrt{3}}{9} \]
Substitute.✓ Proved

Answer \( \frac{17 \sqrt{3}}{9} \)

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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup incorrectly identifies the constant term d as -5 instead of 5. For the plane 3x + 3y + 3z = -5, the standard form is 3x + 3y + 3z + 5 = 0,

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup incorrectly identifies the constant term d as -5 instead of 5. For the plane 3x + 3y + 3z = -5, the standard form is 3x + 3y + 3z + 5 = 0,
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly substitutes x=1 and y=1 instead of x=1 and y=2 for the point (1, 2, 1). Additionally, the formula application is inconsistent with the plane equation 3x + 3y + 3z = -5, where d should be -5, leading to a numerator of |3(1) + 3(2) + 3(1) - (-5)| = 17, but the code shows 5 + (3*1 + 3*1 + 3*2) which implies a sign error or misinterpretation of d as 5 while also using the wrong y-coordinate.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The setup incorrectly identifies the constant term d. For the plane 3x + 3y + 3z = -5, d is -5, so the numerator should be |3(1) + 3(2) + 3(1) - (-5)| = 17. The solution uses |5 + ...| which coincidentally gives the same magnitude but implies a sign error in the formula application (using +5 instead of -(-5) or misidentifying d as 5). While the numerical result is correct, the reasoning in the substitution line is flawed because it treats the constant on the RHS as +5 in a way that suggests the formula is |ax+by+cz+d|/||n|| with d=5, whereas the standard form is ax+by+cz=d, so the term is -d. More critically, the text says "Abs(5 + ...)", implying d=5 was used in a formula like |ax+by+cz+d|, but the plane equation is = -5. If the formula is |ax+by+cz-d|, then -d = 5, so d=-5. The substitution shows 5 + 3+3+6 = 17. This is numerically correct for |3+6+3 - (-5)|, but the notation "5 + ..." is ambiguous and potentially misleading if interpreted as adding the RHS constant directly without sign adjustment. However, the bigger issue is that the sentence defines the formula with 'd' but the substitution uses '5' where '-d' should be if d=-5, or '5' if the formula was |ax+by+cz+d| and the plane was 3x+3y+3z+5=0. The plane is 3x+3y+3z=-5, so 3x+3y+3z+5=0. The formula |ax+by+cz+d|/||n|| with d=5 works. The substitution is Abs(5 + 3(1)+3(2)+3(1)). This is correct. Wait, let me re-read carefully. Plane: 3x + 3y + 3z = -5 => 3x + 3y + 3z + 5 = 0. Formula: |ax0 + by0 + cz0 + d| / ||n|| where d is the constant term in the implicit equation. Here a=3, b=3, c=3, d=5. Numerator: |3(1) + 3(2) + 3(1) + 5| = |3 + 6 + 3 + 5| = 17. Denominator: sqrt(3^2+3^2+3^2) = sqrt(27) = 3*sqrt(3). Result: 17 / (3*sqrt(3)) = 17*sqrt(3)/9. The solution sentence says: Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖. This implies the standard form ax+by+cz=d. If so, d = -5. Numerator should be |3(1)+3(2)+3(1) - (-5)| = |12 + 5| = 17. The substitution line says: Abs(5 + (3*1 + 3*1 + 3*2)). Note: 3*1 + 3*1 + 3*2 = 3 + 3 + 6 = 12. (Wait, y0=2, so 3*2=6. The substitution has 3*1 for y? No, it says 3*1 + 3*1 + 3*2. x=1, y=1?? The point is (1,2,1). So y should be 2. The substitution uses 3*1 for the y-term? Let's check: 3*1 (x) + 3*1 (y?) + 3*2 (z?). If the point is (1,2,1), then 3x=3, 3y=6, 3z=3. Sum = 12. The substitution calculates: 3*1 + 3*1 + 3*2 = 3 + 3 + 6 = 12. This implies y=1. But the point is (1,2,1). So y=2. The substitution used y=1 instead of y=2? Or did it swap y and z? Point: (1, 2, 1). Terms: 3x + 3y + 3z. 3(1) + 3(2) + 3(1) = 3 + 6 + 3 = 12. Substitution in line 2: 3*1 + 3*1 + 3*2. This sums to 12. It seems the solver swapped the values for y and z (using 1 for y and 2 for z) or just wrote them in a confusing order. Since the coefficients are all 3, the sum is the same (12). So the numerical result is correct. However, look at the constant term. Sentence: |ax₀ + by₀ + cz₀ − d|. Substitution: Abs(5 + ...). If the formula is minus d, and the plane is = -5, then d=-5, so -d = 5. So Abs(5 + ...) is correct. Is there an error? The point is (1, 2, 1). The substitution uses 3*1, 3*1, 3*2. This corresponds to x=1, y=1, z=2. The problem states the point is (1, 2, 1). So y=2, z=1. The substitution used y=1, z=2. Because the coefficients for y and z are identical (both 3), the sum 3y+3z is 3(1)+3(2)=9 in the substitution, and 3(2)+3(1)=9 in the correct calculation. The result is numerically correct by coincidence. Is this an error? Yes, substituting the wrong coordinates for y and z is an error, even if the result is correct due to symmetry. A student might think it's okay to swap coordinates. Also, the sentence defines d, but the substitution uses 5. If the student thinks d=5, they are using the form ax+by+cz+d=0. The sentence says ax+by+cz-d. This is a mismatch in notation vs substitution, though numerically consistent if d=-5. The primary error is the substitution of coordinates: y=1 instead of y=2. This is a factual error in the step.
gpt-oss:20b: fail (misleading) 2026-10-18 — The sentence states the distance formula with a minus sign before d, but for the plane 3x+3y+3z+5=0 the correct formula uses +d. The student would learn an incorrect sign convention.

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 2026-10-19 with SymPy 1.14.0.