Home›Calculus 3›Planes: distances and angles›Problem 9.883

Planes: distances and angles

Problem 9.883 · easy

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

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

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

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 dot product calculation in line 2 is incorrect: (-1)(2) + (2)(3) + (-3)(0) = 4, but the line computes -1*2 - 3*0 + 2*3, omitting the z-component of the normal vector in the expression and misordering terms, although the final result 4 is coincidentally correct. More importantly, the sentence in line 1 claims ax+by+cz = n·P, which is correct, but line 2's calculation is presented as n·P without showing the correct vector components being multiplied, potentially confusing the student about which components correspond to which coordinates. Actually, looking closer: n = <-1, 2, -3>, P = <2, 3, 0>. n·P = -1*2 + 2*3 + -3*0 = -6 + 0 = -6? No. -1*2 = -2. 2*3 = 6. -3*0 = 0. Sum = 4. The calculation in line 2 is -1*2 - 3*0 + 2*3. This is -2 - 0 + 6 = 4. The terms are just reordered. It is technically correct arithmetic, but the order is weird. Wait, is it wrong? No. But let's look at the sentence. 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This is correct. The equation is -x + 2y - 3z = 4. This is correct. Why would it fail? Let me re-read the prompt. 'Find what is wrong'. Is there a subtle error? The normal vector is <-1, 2, -3>. The point is (2, 3, 0). Equation: -1(x-2) + 2(y-3) - 3(z-0) = 0 => -x + 2 + 2y - 6 - 3z = 0 => -x + 2y - 3z = 4. The solution is correct. The calculation in line 2 is -1*2 - 3*0 + 2*3. This corresponds to n_x*P_x + n_z*P_z + n_y*P_y. It's just a permutation. It is not an error. However, usually one writes them in order. Is it misleading? Maybe. But is it an error? No. Let me check the 'proved' tag. The equation is marked proved. I should not re-check algebra. But I must check if the sentence models the problem. The sentence says 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This is a standard derivation. The next line calculates n·P. The result is 4. The final answer is -x + 2y - 3z = 4. This is correct. Is there any hidden trap? 'right direction of rotation' - not applicable. 'right region' - not applicable. 'hypotheses hold' - yes. 'conclusion follows' - yes. Maybe the issue is that the solution doesn't explicitly state the final equation? The problem asks for 'an equation'. The stated answer is provided separately. The solution lines are just the derivation. Line 1 sets up the formula. Line 2 calculates the constant. It doesn't write the final equation. Is that a failure? The prompt says 'Review it'. If the solution is incomplete, it might be a style issue or error. But usually, these snippets are parts of a larger solution. However, if this is the *entire* solution, it fails to state the final equation. But the 'Stated answer' is separate. The solution lines are 1 and 2. Line 2 ends with '-- n · P'. It doesn't conclude with the equation. This is likely 'style' or 'misleading' if presented as a complete solution. But often these are steps. Let's look for a harder error. Did I misread the normal vector? <-1, 2, -3>. Point (2, 3, 0). n·P = -2 + 6 + 0 = 4. Correct. What if the sentence 'ax + by + cz = n · P' is considered sloppy because a,b,c are not defined? They are implied by n. This is standard shorthand. I will mark it as pass if it's correct. Wait, look at line 2 again. `-1*2 - 3*0 + 2*3`. The normal vector is `<-1, 2, -3>`. The components are -1, 2, -3. The point is `2, 3, 0`. The dot product is `(-1)(2) + (2)(3) + (-3)(0)`. The line writes `-1*2 - 3*0 + 2*3`. This is `n_x P_x + n_z P_z + n_y P_y`. It is correct. I see no error. I will pass it.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-11-05 — [domain objection, downgraded to style] The dot product calculation in line 2 is incorrect: (-1)(2) + (2)(3) + (-3)(0) = 4, but the line computes -1*2 - 3*0 + 2*3, omitting the z-component of the normal vector in the expression and misordering terms, although the final result 4 is coincidentally correct. More importantly, the sentence in line 1 claims ax+by+cz = n·P, which is correct, but line 2's calculation is presented as n·P without showing the correct vector components being multiplied, potentially confusing the student about which components correspond to which coordinates. Actually, looking closer: n = <-1, 2, -3>, P = <2, 3, 0>. n·P = -1*2 + 2*3 + -3*0 = -6 + 0 = -6? No. -1*2 = -2. 2*3 = 6. -3*0 = 0. Sum = 4. The calculation in line 2 is -1*2 - 3*0 + 2*3. This is -2 - 0 + 6 = 4. The terms are just reordered. It is technically correct arithmetic, but the order is weird. Wait, is it wrong? No. But let's look at the sentence. 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This is correct. The equation is -x + 2y - 3z = 4. This is correct. Why would it fail? Let me re-read the prompt. 'Find what is wrong'. Is there a subtle error? The normal vector is <-1, 2, -3>. The point is (2, 3, 0). Equation: -1(x-2) + 2(y-3) - 3(z-0) = 0 => -x + 2 + 2y - 6 - 3z = 0 => -x + 2y - 3z = 4. The solution is correct. The calculation in line 2 is -1*2 - 3*0 + 2*3. This corresponds to n_x*P_x + n_z*P_z + n_y*P_y. It's just a permutation. It is not an error. However, usually one writes them in order. Is it misleading? Maybe. But is it an error? No. Let me check the 'proved' tag. The equation is marked proved. I should not re-check algebra. But I must check if the sentence models the problem. The sentence says 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This is a standard derivation. The next line calculates n·P. The result is 4. The final answer is -x + 2y - 3z = 4. This is correct. Is there any hidden trap? 'right direction of rotation' - not applicable. 'right region' - not applicable. 'hypotheses hold' - yes. 'conclusion follows' - yes. Maybe the issue is that the solution doesn't explicitly state the final equation? The problem asks for 'an equation'. The stated answer is provided separately. The solution lines are just the derivation. Line 1 sets up the formula. Line 2 calculates the constant. It doesn't write the final equation. Is that a failure? The prompt says 'Review it'. If the solution is incomplete, it might be a style issue or error. But usually, these snippets are parts of a larger solution. However, if this is the *entire* solution, it fails to state the final equation. But the 'Stated answer' is separate. The solution lines are 1 and 2. Line 2 ends with '-- n · P'. It doesn't conclude with the equation. This is likely 'style' or 'misleading' if presented as a complete solution. But often these are steps. Let's look for a harder error. Did I misread the normal vector? <-1, 2, -3>. Point (2, 3, 0). n·P = -2 + 6 + 0 = 4. Correct. What if the sentence 'ax + by + cz = n · P' is considered sloppy because a,b,c are not defined? They are implied by n. This is standard shorthand. I will mark it as pass if it's correct. Wait, look at line 2 again. `-1*2 - 3*0 + 2*3`. The normal vector is `<-1, 2, -3>`. The components are -1, 2, -3. The point is `2, 3, 0`. The dot product is `(-1)(2) + (2)(3) + (-3)(0)`. The line writes `-1*2 - 3*0 + 2*3`. This is `n_x P_x + n_z P_z + n_y P_y`. It is correct. I see no error. I will pass it.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: fail (error) 2026-11-04 — The dot product calculation in line 2 is incorrect: the normal vector is <-1, 2, -3>, so n · P should be (-1)(2) + (2)(3) + (-3)(0) = 4, but the line writes '-1*2 - 3*0 + 2*3', which incorrectly associates the -3 component with the z-coordinate of P (which is 0) and the 2 component with the y-coordinate, but the order of terms is jumbled and implies a mistake in component matching if interpreted strictly as <a,b,c> dot <x,y,z>. Wait, let's look closer. n = <-1, 2, -3>. P = <2, 3, 0>. n·P = -1*2 + 2*3 + -3*0 = -6 + 6 = 0. The solution claims the result is 4. The calculation shown is -1*2 - 3*0 + 2*3. This evaluates to -2 + 0 + 6 = 4. The arithmetic is correct for the terms written, but the terms written do not match n·P. The term '-3*0' corresponds to the z-component of n (-3) and z-component of P (0). The term '2*3' corresponds to y-component of n (2) and y-component of P (3). The term '-1*2' corresponds to x-component of n (-1) and x-component of P (2). So the calculation IS n·P. Why did I think it was 0? -1*2 = -2. 2*3 = 6. -3*0 = 0. Sum = 4. Okay, the calculation is actually correct. Let me re-read the normal vector. <-1, 2, -3>. Point (2, 3, 0). -1(2) + 2(3) - 3(0) = -2 + 6 - 0 = 4. The equation is -x + 2y - 3z = 4. The solution is correct. Why did I flag it? I misread the vector or point in my head. Let me double check the 'sentence' part. 'n · (r − P) = 0, i.e. ax + by + cz = n · P.' This is standard. The equation line is marked 'proved'. The sentence is correct. The calculation is correct. The final answer matches. It seems sound.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: fail (error) 2026-11-04 — The calculation in line 2 is incorrect; it computes -1*2 + 2*3 = 4 but omits the z-component (-3*0) in the text, and more importantly, the dot product < -1, 2, -3 > · < 2, 3, 0 > is actually -2 + 6 + 0 = 4, so the arithmetic is coincidentally correct but the expression written '-1*2 - 3*0 + 2*3' is a jumbled representation of the dot product that could confuse a student about the order of operations or component pairing. However, the primary issue is that line 2 is marked 'unchecked' but contains arithmetic that should be verified, and the phrasing 'n . P' implies the dot product, but the written sum '-1*2 - 3*0 + 2*3' is not the standard order (usually x, y, z) and mixes terms. Wait, -1*2 is x, -3*0 is z, 2*3 is y. It's just out of order. The result 4 is correct. The final equation -x + 2y - 3z = 4 is correct. Is there an error? The prompt says 'find what is wrong'. The solution is actually correct. Let me re-read carefully. Line 1 says 'ax + by + cz = n . P'. This is a valid scalar equation form. Line 2 calculates n . P. The calculation is -1*2 + 2*3 + (-3)*0 = 4. The text writes '-1*2 - 3*0 + 2*3'. This equals -2 - 0 + 6 = 4. The arithmetic is correct. The final answer is correct. Why would it fail? Perhaps the 'unchecked' tag is the issue? No, I am reviewing the sentences. The sentence in line 1 is a valid setup. The equation in line 2 is arithmetically correct. The final answer matches. I will mark it pass.
gpt-oss:20b: pass 2026-11-04

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 2026-11-05 with SymPy 1.14.0.