Home›Calculus 3›Planes: distances and angles›Problem 9.771

Planes: distances and angles

Problem 9.771 · easy

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

n · (r − P) = 0, i.e. ax + by + cz = n · P.
\[ 2 \left(-1\right) + 1 \cdot 3 - 2 \left(-2\right) = 5 \]
n · P.✓ Proved

Answer \( x + 2 y - 2 z = 5 \)

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 (style) — [domain objection, downgraded to style] 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 = ax_0 + by_0 + cz_0. The solution writes ax + by + cz = n · P, but P is a point, not a vector, so the dot product n · P is undefined notation. Furthermore, the calculation in line 2 computes 1*3 + 2*(-1) - 2*(-2) = 5, but the text labels it as 'n · P' while computing components in a mixed order (2*(-1) + 1*3...) that doesn't match the standard dot product definition <1,2,-2>·<3,-1,-2> = 1*3 + 2*(-1) + (-2)*(-2). While the numerical result 5 is correct, the reasoning and notation are fundamentally flawed.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-19 — [domain objection, downgraded to style] 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 = ax_0 + by_0 + cz_0. The solution writes ax + by + cz = n · P, but P is a point, not a vector, so the dot product n · P is undefined notation. Furthermore, the calculation in line 2 computes 1*3 + 2*(-1) - 2*(-2) = 5, but the text labels it as 'n · P' while computing components in a mixed order (2*(-1) + 1*3...) that doesn't match the standard dot product definition <1,2,-2>·<3,-1,-2> = 1*3 + 2*(-1) + (-2)*(-2). While the numerical result 5 is correct, the reasoning and notation are fundamentally flawed.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The setup in line 1 is incorrect because it swaps the coefficients of x and y relative to the given normal vector <1, 2, -2>. The equation should be 1x + 2y - 2z = d, but the text implies ax+by+cz corresponds to the normal components in a way that leads to the calculation in line 2 using 2 for the y-component and 1 for the x-component, which is actually correct for the normal <1, 2, -2> if a=1, b=2. Wait, let's look closer. Line 1 says 'ax + by + cz = n . P'. Line 2 calculates '2*(-1) + 1*3 - 2*(-2)'. This calculation uses 2 for the y-term and 1 for the x-term. The normal is <1, 2, -2>. So a=1, b=2, c=-2. The calculation is 1*(3) + 2*(-1) + (-2)*(-2) = 3 - 2 + 4 = 5. The calculation in line 2 is written as '2*(-1) + 1*3 - 2*(-2)'. This is 2y + 1x - 2z. This matches the normal <1, 2, -2> if we associate 1 with x and 2 with y. The sentence in line 1 is vague ('ax+by+cz') but doesn't explicitly assign values. However, the calculation in line 2 is correct for the dot product. The final answer x + 2y - 2z = 5 is correct. Is there an error? The sentence says 'n . (r - P) = 0, i.e. ax + by + cz = n . P'. This is standard. The calculation is correct. The result is correct. Why would it fail? Let's re-read carefully. Normal <1, 2, -2>. Point (3, -1, -2). Dot product: 1*3 + 2*(-1) + (-2)*(-2) = 3 - 2 + 4 = 5. Equation: x + 2y - 2z = 5. The solution provides this. The calculation line writes '2*(-1) + 1*3 - 2*(-2)'. This is 2y + 1x - 2z. This is equivalent to 1x + 2y - 2z. The order of terms in the sum doesn't matter. The sentence is generic. The equation is proved. The answer is stated. It seems correct. Wait, look at line 1 again. 'n . (r - P) = 0, i.e. ax + by + cz = n . P'. This implies the plane equation is ax+by+cz=d. It does not explicitly state a=1, b=2, c=-2. But line 2 computes the RHS. It doesn't write the final equation explicitly in the steps, but the 'Stated answer' is separate. The prompt asks to review the solution lines. Line 1 is a general formula. Line 2 is a calculation. The connection is implicit. Is it misleading? A student might not know which a,b,c to use if not stated. But line 2 uses the correct numbers. It's a bit sparse but not wrong. Let's look for a subtle error. 'n . (r - P) = 0'. r = <x,y,z>. P = <3,-1,-2>. n = <1,2,-2>. n . r = n . P. 1x + 2y - 2z = 5. The solution is correct. I will pass it.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] 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 = d where d = ax_0 + by_0 + cz_0. The solution claims the equation is ax + by + cz = n · P, but n · P is a scalar value (5), not a vector equation, and the left side 'ax+by+cz' implies using the components of n as coefficients a,b,c. While the final numerical answer is correct, the stated formula 'n · (r − P) = 0, i.e. ax + by + cz = n · P' is notationally confused and mathematically imprecise because 'n · P' is a scalar while the left side of the 'i.e.' clause suggests a vector operation or is missing the variable vector r. More critically, line 1 presents 'ax + by + cz = n · P' as the general form derived from n · (r - P) = 0. This derivation skips the step of expanding n · r - n · P = 0 to n · r = n · P. If a, b, c are components of n, then ax+by+cz is n·r. So n·r = n·P is correct. However, the notation 'ax+by+cz' usually implies a, b, c are constants. The sentence is technically correct if interpreted loosely, but let's look closer. Is it wrong? n · (r - P) = 0 => n · r = n · P. If n = <a,b,c> and r = <x,y,z>, then ax+by+cz = n·P. This is correct. Why did I think it was an error? Let's re-read carefully. 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This is a valid derivation. The calculation in line 2 is 2*(-1) + 1*3 - 2*(-2) = -2 + 3 + 4 = 5. This is n · P. The final equation is x + 2y - 2z = 5. This matches the stated answer. Is there a subtle error? The problem asks for 'an equation'. The solution provides one. The setup is standard. Wait, look at line 1 again. 'ax + by + cz = n · P'. Here a,b,c are implicitly the components of n. This is standard shorthand. Is it misleading? No. Is it an error? No. Let me check the calculation again. n = <1, 2, -2>. P = (3, -1, -2). n · P = 1(3) + 2(-1) + (-2)(-2) = 3 - 2 + 4 = 5. Line 2 calculates 2*(-1) + 1*3 - 2*(-2). This is 2y_0 + 1x_0 - 2z_0. The order of terms in the dot product is commutative, so 1*3 + 2*(-1) + (-2)*(-2) is the same as 2*(-1) + 1*3 - 2*(-2). The calculation is correct. The result is correct. The setup is correct. Why would this fail? Perhaps the 'i.e.' is too loose? No, it's standard. Let's look for a 'misleading' aspect. The sentence says 'ax + by + cz = n · P'. It does not explicitly define a, b, c as the components of n. A student might think a, b, c are arbitrary. However, in context of 'n · (r-P)=0', it's implied. Is this 'style' or 'misleading'? It's slightly imprecise but not false. Let's look at the prompt requirements again. 'Check that... each conclusion follows... nothing would teach a student something false.' If a student reads 'ax+by+cz = n·P' and doesn't know a,b,c come from n, they might be confused. But usually a,b,c are defined as normal components. Let's assume this is acceptable. Is there a sign error? n = <1, 2, -2>. Equation: 1x + 2y - 2z = 5. Point (3, -1, -2): 3 - 2 + 4 = 5. Correct. I will mark this as pass.
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 2026-10-19 with SymPy 1.14.0.