Home›Calculus 3›Planes: distances and angles›Problem 9.639

Planes: distances and angles

Problem 9.639 · easy

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

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

Answer \( 3 x + 3 y - z = -5 \)

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 (error) — The dot product calculation in line 2 is incorrect: 3*(-3) + 3*2 + (-1)*2 = -9 + 6 - 2 = -5, but the solution writes 3*(-3) - 1*2 + 3*2, which omits the 3y term's contribution to the dot product logic or misorders terms confusingly. More critically, the formula stated in line 1 'ax + by + cz = n · P' is ambiguous without defining P as the position vector of the point. The calculation 3*(-3) - 1*2 + 3*2 evaluates to -9 -2 + 6 = -5, which happens to be the correct constant, but the term ordering suggests a confusion about which component multiplies which coordinate (specifically, it looks like it multiplied z-component -1 by y-coordinate 2, and y-component 3 by z-coordinate 2). While the final number is right, the intermediate expression is misleadingly written as if n=<3, -1, 3> or P=<-3, 2, 2> was dotted incorrectly. Actually, looking closer: n=<3,3,-1>, P=<-3,2,2>. n.P = 3(-3) + 3(2) + (-1)(2) = -9 + 6 - 2 = -5. The solution writes: 3*(-3) - 1*2 + 3*2. This is 3*x + (-1)*y + 3*z? No, it's just a jumbled sum. It's not strictly an 'error' in the final result, but the representation of the dot product is sloppy. However, the prompt asks to find what is wrong. Is it wrong? The equation 3x+3y-z=-5 is correct. The sentence 'n . (r - P) = 0, i.e. ax + by + cz = n . P' is standard. The calculation line is just a calculation. Is it 'wrong'? It calculates the correct value. But wait, look at the terms: 3*(-3) is n_x * P_x. -1*2 is n_z * P_y? No, n_z is -1, P_y is 2. So it multiplied n_z by P_y. And 3*2 is n_y * P_z? n_y is 3, P_z is 2. So it calculated n_x P_x + n_z P_y + n_y P_z. This is not the dot product n.P. It is a coincidence that n_z P_y + n_y P_z = (-1)(2) + (3)(2) = 4 and n_y P_y + n_z P_z = (3)(2) + (-1)(2) = 4. So the result is correct by coincidence, but the method shown is wrong. A student would learn that you can mix up the components of the normal vector and point vector as long as you use the same numbers? No, they would learn that the dot product is commutative in a way that allows swapping y and z components if their values happen to make the sum equal? No, they would just be confused. This is a 'misleading' or 'error' in the derivation step. Since the step claims to calculate n.P but calculates something else (that happens to equal n.P), it is an error in the solution's logic.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The dot product calculation in line 2 is incorrect: 3*(-3) + 3*2 + (-1)*2 = -9 + 6 - 2 = -5, but the solution writes 3*(-3) - 1*2 + 3*2, which omits the 3y term's contribution to the dot product logic or misorders terms confusingly. More critically, the formula stated in line 1 'ax + by + cz = n · P' is ambiguous without defining P as the position vector of the point. The calculation 3*(-3) - 1*2 + 3*2 evaluates to -9 -2 + 6 = -5, which happens to be the correct constant, but the term ordering suggests a confusion about which component multiplies which coordinate (specifically, it looks like it multiplied z-component -1 by y-coordinate 2, and y-component 3 by z-coordinate 2). While the final number is right, the intermediate expression is misleadingly written as if n=<3, -1, 3> or P=<-3, 2, 2> was dotted incorrectly. Actually, looking closer: n=<3,3,-1>, P=<-3,2,2>. n.P = 3(-3) + 3(2) + (-1)(2) = -9 + 6 - 2 = -5. The solution writes: 3*(-3) - 1*2 + 3*2. This is 3*x + (-1)*y + 3*z? No, it's just a jumbled sum. It's not strictly an 'error' in the final result, but the representation of the dot product is sloppy. However, the prompt asks to find what is wrong. Is it wrong? The equation 3x+3y-z=-5 is correct. The sentence 'n . (r - P) = 0, i.e. ax + by + cz = n . P' is standard. The calculation line is just a calculation. Is it 'wrong'? It calculates the correct value. But wait, look at the terms: 3*(-3) is n_x * P_x. -1*2 is n_z * P_y? No, n_z is -1, P_y is 2. So it multiplied n_z by P_y. And 3*2 is n_y * P_z? n_y is 3, P_z is 2. So it calculated n_x P_x + n_z P_y + n_y P_z. This is not the dot product n.P. It is a coincidence that n_z P_y + n_y P_z = (-1)(2) + (3)(2) = 4 and n_y P_y + n_z P_z = (3)(2) + (-1)(2) = 4. So the result is correct by coincidence, but the method shown is wrong. A student would learn that you can mix up the components of the normal vector and point vector as long as you use the same numbers? No, they would learn that the dot product is commutative in a way that allows swapping y and z components if their values happen to make the sum equal? No, they would just be confused. This is a 'misleading' or 'error' in the derivation step. Since the step claims to calculate n.P but calculates something else (that happens to equal n.P), it is an error in the solution's logic.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The dot product calculation in line 2 is incorrect; it computes 3(-3) + (-1)(2) + 3(2) = -5, but the normal vector is <3, 3, -1>, so the y-component should be 3*2, not -1*2. The correct dot product is 3(-3) + 3(2) + (-1)(2) = -5, which happens to yield the same result by coincidence, but the setup implies the wrong normal vector components were used (specifically swapping the y and z components of the normal vector in the calculation). Wait, let's re-read carefully. Normal is <3, 3, -1>. Point is (-3, 2, 2). Dot product: 3*(-3) + 3*(2) + (-1)*(2) = -9 + 6 - 2 = -5. The solution line says: 3*(-3) - 1*2 + 3*2. This corresponds to normal <3, -1, 3> or <3, 3, -1> with terms reordered? 3*(-3) is x. -1*2 is z? 3*2 is y? If the order is x, z, y, then it matches <3, 3, -1> if we map 3->x, 3->y, -1->z. The line writes 3*(-3) [x] - 1*2 [z?] + 3*2 [y?]. It seems to have calculated 3x + 3y - z but wrote the terms in the order x, z, y? Or did it use normal <3, -1, 3>? If it used <3, -1, 3>, the plane equation would be 3x - y + 3z = -5. The stated answer is 3x + 3y - z = -5. The calculation 3*(-3) - 1*2 + 3*2 = -9 - 2 + 6 = -5. This calculation matches the coefficients 3, -1, 3 applied to -3, 2, 2. But the normal is <3, 3, -1>. The correct calculation is 3*(-3) + 3*(2) + (-1)*(2) = -9 + 6 - 2 = -5. The solution's calculation is arithmetically correct for the numbers written, but the numbers written correspond to a different normal vector (<3, -1, 3>) or a jumbled order. However, since the final answer 3x + 3y - z = -5 is correct, the error is in the intermediate explanation. The sentence says 'n . P' and then shows a calculation that doesn't match n=<3,3,-1> directly unless reordered. It is misleading/error because it suggests the dot product of <3,3,-1> and <-3,2,2> is calculated as 3(-3) - 1(2) + 3(2), which is not the standard expansion and looks like a mistake in identifying components.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The dot product calculation in line 2 is incorrect: 3*(-3) + 3*2 + (-1)*2 equals -5, but the line writes 3*(-3) - 1*2 + 3*2, omitting the 3y term and misordering the components. While the final constant -5 is coincidentally correct, the displayed arithmetic is wrong.
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.