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Planes: distances and angles

Problem 9.716 · easy

Find an equation of the plane through \( \displaystyle (-3, -1, -1) \) with normal vector \( \displaystyle \left\langle -4, 3, -3 \right\rangle \).

n · (r − P) = 0, i.e. ax + by + cz = n · P.
\[ 3 \left(-1\right) - 3 \left(-1\right) - 4 \left(-3\right) = 12 \]
n · P.✓ Proved

Answer \( - 4 x + 3 y - 3 z = 12 \)

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 setup in line 1 incorrectly states the plane equation as ax + by + cz = n · P, which implies the plane passes through the origin or misidentifies the constant term. The correct scalar equation is a(x - x0) + b(y - y0) + c(z - z0) = 0, which simplifies to ax + by + cz = d where d = a*x0 + b*y0 + c*z0. While line 2 correctly calculates d = 12, the formula presented in line 1 is mathematically incorrect and misleading.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The setup in line 1 incorrectly states the plane equation as ax + by + cz = n · P, which implies the plane passes through the origin or misidentifies the constant term. The correct scalar equation is a(x - x0) + b(y - y0) + c(z - z0) = 0, which simplifies to ax + by + cz = d where d = a*x0 + b*y0 + c*z0. While line 2 correctly calculates d = 12, the formula presented in line 1 is mathematically incorrect and misleading.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The setup in line 1 claims the plane equation is ax + by + cz = n · P. This is incorrect; the correct scalar equation is a(x-x_0) + b(y-y_0) + c(z-z_0) = 0, which simplifies to ax + by + cz = ax_0 + by_0 + cz_0. While the right-hand side value (12) happens to be correct for the given point and normal, the formula stated implies that the constant term is always the dot product of the normal vector with the position vector of the point, which is only true if the equation is written as n · r = n · P. However, the left side of the stated equation 'ax + by + cz' usually implies coefficients a,b,c from the normal vector <a,b,c>. The statement 'ax + by + cz = n · P' is ambiguous and potentially misleading if interpreted as the general form without specifying that a,b,c are the components of n. More critically, line 1 presents a formula that is not generally true for any plane equation unless specifically defined. The standard point-normal form is n · (r - P) = 0. Expanding this gives n · r = n · P. If a,b,c are components of n, then ax+by+cz = n·P is correct. Let's re-read carefully. 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This step assumes a,b,c are the components of n. This is a standard convention. Is it wrong? No. Is it misleading? It's a bit loose but standard. Let's look at the calculation. n = <-4, 3, -3>, P = (-3, -1, -1). n·P = (-4)(-3) + (3)(-1) + (-3)(-1) = 12 - 3 + 3 = 12. The equation is -4x + 3y - 3z = 12. The stated answer matches. The logic holds. Why did I think it was an error? The sentence says 'ax + by + cz = n · P'. It doesn't explicitly define a,b,c as components of n, but it's implied by the context of 'normal vector <a,b,c>'. This is acceptable shorthand. Wait, looking at the previous turn's instructions: 'Check that... each conclusion follows from the lines before it'. Line 1 states a formula. Line 2 calculates the RHS. The final answer is not explicitly derived in a line, but the 'Stated answer' is provided. The solution block only has 2 lines. Line 1 is a sentence. Line 2 is an equation. The solution is incomplete because it doesn't explicitly write the final equation -4x + 3y - 3z = 12 as a derived step, but rather just calculates the constant. However, the prompt asks to review the *provided* solution lines. Line 1 is a valid setup. Line 2 is a valid calculation. The connection to the final answer is implicit. Is there a subtle error? 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This is mathematically correct if a,b,c are components of n. The severity 'error' might be too strong if it's just implicit. But wait, look at the normal vector in the problem: <-4, 3, -3>. In line 1, it uses generic a,b,c. In line 2, it calculates n·P. The solution is technically sound but very brief. Let's look for a real error. Ah, the problem asks for 'an equation'. The solution provides the constant. It doesn't write the full equation. But usually, these reviews focus on correctness of steps. Is there a sign error? n = <-4, 3, -3>. P = (-3, -1, -1). Dot product: 12 - 3 + 3 = 12. Correct. Equation: -4x + 3y - 3z = 12. Correct. Why would this fail? Maybe the sentence 'ax + by + cz = n · P' is considered 'misleading' because it doesn't specify that a,b,c are the components of n? Or perhaps the 'i.e.' implies equivalence which requires defining a,b,c. Without defining a,b,c, the statement is ambiguous. Given the adversarial nature, this ambiguity is a flaw. However, is it an 'error'? A student might think a,b,c are arbitrary. It's better to say 'Let n = <a,b,c>'. I will mark it as 'misleading' or 'style'. But wait, look at line 2: '3*(-1) - 3*(-1) - 4*(-3) = 12'. The order of terms in the dot product calculation is 3*(-1) [y-component] then -3*(-1) [z-component] then -4*(-3) [x-component]. This is just a sum, order doesn't matter. The calculation is correct. The main issue is the lack of explicit definition of a,b,c in line 1. I will call it 'style' or 'misleading'. Let's look closer. Is there a more fundamental error? No. The solution is correct. I will pass it? No, the prompt says 'find what is wrong'. If it's correct, say pass. Is it correct? Yes. The shorthand is standard in calculus contexts. I will pass.
gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution stops after computing \(n\cdot P\) but never writes the resulting plane equation \( -4x+3y-3z=12\). The first sentence also misstates the rel
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the point-normal form of the plane equation. The calculation of the constant term (n · P) is correct, and the resulting equation matches the stated answer.
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.