Home›Calculus 3›Planes: distances and angles›Problem 9.812

Planes: distances and angles

Problem 9.812 · easy

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

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

Answer \( 2 x + 4 y + z = -1 \)

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 is incorrect. The standard scalar equation of a plane 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. The solution incorrectly states the formula as ax + by + cz = n · P without accounting for the variable components (x, y, z) being multiplied by the normal vector components in the dot product n · r. Specifically, it implies the left side is just the coefficients, missing the variables x, y, z entirely in the formula description, or conflates the dot product n·r with just n. While the numerical calculation in line 2 is correct for the RHS, the formula stated in line 1 is mathematically nonsensical as written (ax+by+cz is not equal to n·P generally; ax+by+cz = d where d=n·P). A clearer statement would be 'The equation is n · r = n · P' or 'a(x-x_0) + ... = 0'. As written, it suggests the equation of the plane is just the constants, which is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The setup in line 1 is incorrect. The standard scalar equation of a plane 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. The solution incorrectly states the formula as ax + by + cz = n · P without accounting for the variable components (x, y, z) being multiplied by the normal vector components in the dot product n · r. Specifically, it implies the left side is just the coefficients, missing the variables x, y, z entirely in the formula description, or conflates the dot product n·r with just n. While the numerical calculation in line 2 is correct for the RHS, the formula stated in line 1 is mathematically nonsensical as written (ax+by+cz is not equal to n·P generally; ax+by+cz = d where d=n·P). A clearer statement would be 'The equation is n · r = n · P' or 'a(x-x_0) + ... = 0'. As written, it suggests the equation of the plane is just the constants, which is wrong.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The setup in line 1 is mathematically incorrect. The standard scalar equation of a plane is a(x-x_0) + b(y-y_0) + c(z-z_0) = 0, which expands to ax + by + cz = ax_0 + by_0 + cz_0. The solution incorrectly states the formula as ax + by + cz = n · P without subtracting the point coordinates from the variables, implying the plane passes through the origin or misidentifying the constant term's derivation. While the numerical calculation in line 2 is correct for the constant term, the logical link in line 1 is flawed.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The setup in line 1 is incorrect because it omits the normal vector components (a, b, c) from the left-hand side of the equation. It claims 'ax + by + cz = n · P' but writes 'n · (r - P) = 0' which expands to 'ax + by + cz = n · P', yet the notation 'n · (r - P) = 0' implies the dot product of the normal vector with the position vector difference, which is correct, but the text 'ax + by + cz = n · P' is presented as the definition without defining a, b, c as the components of n. More critically, line 2 calculates n · P correctly as -1, but the final equation 2x + 4y + z = -1 is derived from the correct normal vector <2, 4, 1>. However, the sentence in line 1 is confusingly written. It says 'n · (r - P) = 0, i.e. ax + by + cz = n · P'. This is technically correct if a,b,c are components of n. But usually, one writes a(x-x0) + ... = 0. The bigger issue is that the solution does not explicitly state the final equation in the provided lines, only the calculation of the constant. Wait, the problem asks for the review of the provided solution. The solution provided is just two lines. Line 1 is a sentence. Line 2 is an equation. The 'Stated answer' is separate. The solution lines provided do not actually produce the final equation '2x + 4y + z = -1'. They only calculate the RHS. A student reading this might not know how to form the LHS. Furthermore, line 1 says 'ax + by + cz = n · P'. It does not define a, b, c. It assumes the reader knows a,b,c come from n. This is a style/clarity issue, but is it an error? The prompt says 'Find what is wrong'. The solution is incomplete. It doesn't state the final answer. But usually, these reviews check the logic of the steps provided. Step 1 is a valid identity. Step 2 is a valid calculation. But Step 1 is misleading because it doesn't specify that a,b,c are the components of n. Also, the standard form is often a(x-x0)+...=0. The form ax+by+cz=d is also standard. The main error is that the solution does not conclude with the equation. However, looking at the 'Stated answer', it seems the solution is meant to lead to it. Let's look closer at Line 1. 'n · (r - P) = 0, i.e. ax + by + cz = n · P'. This is mathematically correct. Line 2 calculates n · P. The missing link is writing the LHS. Is this an error? It's an incomplete solution. But is it 'wrong'? The prompt asks to review the solution. If the solution is just these two lines, it fails to answer the problem. That is an error. Also, Line 1 is slightly ambiguous about where a,b,c come from. I will mark it as error because the solution is incomplete and does not state the final equation.
gpt-oss:20b: pass 2026-10-18

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.