Home›Calculus 3›Planes: distances and angles›Problem 9.638

Planes: distances and angles

Problem 9.638 · easy

Find an equation of the plane through \( \displaystyle (0, -1, -2) \) with normal vector \( \displaystyle \left\langle -3, 4, 2 \right\rangle \).

n · (r − P) = 0, i.e. ax + by + cz = n · P.
\[ 4 \left(-1\right) + 2 \left(-2\right) - 3 \cdot 0 = -8 \]
n · P.✓ Proved

Answer \( - 3 x + 4 y + 2 z = -8 \)

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 sentence claims the plane equation is ax + by + cz = n · P, but the normal vector components are (-3, 4, 2), so the equation should be -3x + 4y + 2z = n · P. The sentence omits the negative sign on the x-term, implying a normal of <3, 4, 2> or just failing to specify the coefficients correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The sentence claims the plane equation is ax + by + cz = n · P, but the normal vector components are (-3, 4, 2), so the equation should be -3x + 4y + 2z = n · P. The sentence omits the negative sign on the x-term, implying a normal of <3, 4, 2> or just failing to specify the coefficients correctly.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the plane equation formula and computes the constant term accurately. The final equation matches the stated answer.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The setup in line 1 is incorrect. The standard plane equation is a(x-x_0) + b(y-y_0) + c(z-z_0) = 0, which simplifies to ax + by + cz = d where d = ax_0 + by_0 + cz_0. The solution claims the right-hand side is n · P, but calculates n · P as -8 in line 2. However, the normal vector is <-3, 4, 2> and P is (0, -1, -2). The dot product is (-3)(0) + (4)(-1) + (2)(-2) = -8. Wait, the equation in line 1 says 'ax + by + cz = n · P'. If a=-3, b=4, c=2, then -3x + 4y + 2z = -8. This matches the stated answer. Let me re-read carefully. Line 1 says 'n · (r - P) = 0, i.e. ax + by + cz = n · P'. This is mathematically incorrect notation/logic. n · (r - P) = 0 implies n · r = n · P. If r = <x,y,z> and n = <a,b,c>, then ax+by+cz = a*x_0 + b*y_0 + c*z_0. The text says 'ax + by + cz = n · P'. This is actually correct if P is the position vector of the point. Let's check the calculation in line 2. 4*(-1) + 2*(-2) - 3*0 = -4 - 4 - 0 = -8. This is correct. The final answer is -3x + 4y + 2z = -8. This is correct. Why would it fail? Ah, look at line 1 again. 'n · (r - P) = 0, i.e. ax + by + cz = n · P'. This implies that the left side is just the components of n dotted with r, but written as 'ax+by+cz'. Usually, the equation is a(x-x_0)... The statement 'ax + by + cz = n · P' is a valid simplification of n · r = n · P. Is there an error? Let's check the vector components. n = <-3, 4, 2>. P = (0, -1, -2). n · P = -8. Equation: -3x + 4y + 2z = -8. The solution is correct. Wait, is there a subtle error? 'n · (r - P) = 0' is the vector form. 'ax + by + cz = n · P' is the scalar form. This is correct. Let me look for 'misleading' or 'style'. The prompt asks to find what is wrong. If it's correct, I should say pass. But usually these adversarial reviews have a trap. Let's look at line 2: '4*(-1) + 2*(-2) - 3*0 = -8'. The order of terms in the dot product calculation is y-component, z-component, x-component. This is fine. Is the normal vector applied correctly in the final answer? The stated answer is '- 3 x + 4 y + 2 z = -8'. The solution doesn't explicitly write the final equation, but line 1 sets up the form and line 2 calculates the constant. The 'Stated answer' is provided in the problem description, not the solution lines. The solution lines are 1 and 2. Line 1 is a sentence. Line 2 is an equation. The solution is incomplete? It doesn't state the final equation. But the prompt says 'Stated answer: ...'. The solution lines are just the derivation. Is the derivation sound? Yes. Is it misleading? No. Is it an error? No. Wait, look at line 1: 'n · (r - P) = 0, i.e. ax + by + cz = n · P'. This assumes r = <x,y,z> and n = <a,b,c>. This is standard. However, strictly speaking, n · r = ax + by + cz. So n · r = n · P becomes ax + by + cz = n · P. This is correct. Why would I fail it? Maybe the variable names? 'ax + by + cz' usually implies a,b,c are constants. They are. Okay, I will pass it. Wait, let me double check the 'adversarial' nature. Is there a sign error? n = <-3, 4, 2>. P = (0, -1, -2). Dot product: 0 - 4 - 4 = -8. Equation: -3x + 4y + 2z = -8. Correct. What if the point was (0, 1, 2)? No, it's (0, -1, -2). What if the normal was <3, -4, -2>? No. Okay, I'll pass. Actually, looking at similar examples, often the 'sentence' is considered 'style' if it's sloppy. 'i.e.' is strong. Is it exactly equivalent? Yes. I will pass.
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.