Home›Calculus 3›Equation of a plane›Problem 9.156

Equation of a plane

Problem 9.156 · hard

Find an equation of the plane through the points \( \displaystyle (-3, -2, 2),\ (2, 0, 0),\ (2, 3, -1) \).

\[ \left[\begin{matrix}5\\2\\-2\end{matrix}\right] \]
PQ.✓ Proved
\[ \left[\begin{matrix}5\\5\\-3\end{matrix}\right] \]
PR.✓ Proved
\[ \left[\begin{matrix}4\\5\\15\end{matrix}\right] \]
A normal vector n = PQ × PR (scaled).✓ Proved
\[ 8 \]
n · P gives the constant.✓ Proved
The plane is 4*x + 5*y + 15*z = 8.

Answer \( 4 x + 5 y + 15 z = 8 \)

Lines: 4 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.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	all three points satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer returned a non-object

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer returned a non-object
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (style) 2026-09-28 — [domain objection, downgraded to style] The vectors PQ and PR are calculated incorrectly. For P=(-3,-2,2), Q=(2,0,0), R=(2,3,-1), PQ should be <5, 2, -2> and PR should be <5, 5, -3>. While the solution lists these correct vectors, the cross product calculation in step 3 is presented as a 'scaled' normal without justification, and more importantly, the dot product in step 4 is ambiguous. Using P=(-3,-2,2) with n=<4,5,15> gives 4(-3)+5(-2)+15(2) = -12-10+30 = 8, which matches. However, the solution text is extremely sparse and the 'unchecked' equations suggest the algebraic steps were skipped or hidden, making it impossible to verify the cross product logic properly. But wait, let's re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. The equations are marked 'unchecked' but the prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. Here they are marked 'unchecked'. This implies I should check them? Or does 'unchecked' mean the CAS didn't check them? The prompt says 'do not re-check the algebra of a line marked proved'. These are marked 'unchecked'. This is ambiguous. However, looking at the vectors: PQ = Q-P = <2-(-3), 0-(-2), 0-2> = <5, 2, -2>. PR = R-P = <2-(-3), 3-(-2), -1-2> = <5, 5, -3>. The cross product PQ x PR = <(2)(-3)-(-2)(5), (-2)(5)-(5)(-3), (5)(5)-(2)(5)> = <-6+10, -10+15, 25-10> = <4, 5, 15>. The normal vector is correct. The constant d = n.P = 4(-3)+5(-2)+15(2) = 8. The equation is correct. Why would it fail? Let's look closer. Step 1 and 2 define the vectors. Step 3 defines the normal. Step 4 defines the constant. Step 5 states the equation. The solution is mathematically correct. Is there a style issue? The vectors are defined as matrices. The cross product is just stated. The dot product is just stated. The prompt asks to review 'sentences'. Step 5 is the only sentence. 'The plane is 4*x + 5*y + 15*z = 8.' This sentence is correct. The previous lines are equations. If the equations are correct (which they are), and the sentence is correct, it should pass. Wait, look at Step 1 and 2. They show `Matrix([...]), Matrix([...])`. This is not an equation. It's two matrices. The label says `-- PQ.`. This is a definition. Is it an equation? No. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Lines 1-4 are labeled `[equation, unchecked]`. But they are not equations. They are assignments or definitions. This is a formatting/style error in the solution representation, but does it make the math wrong? No. However, usually 'unchecked' means I should check it. If I check it, the math is right. Let's look for a subtle error. P=(-3,-2,2). Q=(2,0,0). R=(2,3,-1). Normal <4,5,15>. 4x+5y+15z=8. Check Q: 4(2)+0+0=8. Correct. Check R: 4(2)+5(3)+15(-1)=8+15-15=8. Correct. The solution is correct. Why would it be a fail? Maybe the 'unchecked' tag implies the CAS didn't verify it, so I must? I did, and it's correct. Maybe the issue is that the solution doesn't explicitly state the formula for the plane? Step 5 just states the result. It's a bit brief, but not 'misleading' or 'error'. Let's look at the vectors again. PQ = <5, 2, -2>. PR = <5, 5, -3>. Cross product: i(2*-3 - -2*5) = i(-6+10)=4i. j(5*-3 - 5*-2) = j(-15+10)=-5j? Wait. The middle term of cross product is -(a1*b3 - a3*b1) or a3*b1 - a1*b3. Let's use determinant method. |i j k; 5 2 -2; 5 5 -3|. i(2*-3 - -2*5) = -6+10=4. j(5*-3 - 5*-2) = -15+10=-5. So -j term is -(-5)=5? No. The component is -(5*-3 - 5*-2) = -(-15+10) = -(-5) = 5. Or use cyclic: k(5*5 - 5*2) = 25-10=15. So <4, 5, 15>. The calculation in step 3 is correct. The solution is sound. I will mark it pass.
gpt-oss:20b: pass 2026-09-28

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/plane_through_points, checked 2026-09-28 with SymPy 1.14.0.