Home›Calculus 3›Planes: distances and angles›Problem 9.713

Planes: distances and angles

Problem 9.713 · easy

Find the distance from \( \displaystyle (3, -1, -2) \) to the plane \( \displaystyle - x + 3 y + 2 z = -1 \).

Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖.
Reviewed
\[ \frac{\left|{2 \left(-2\right) - 1 \cdot 3 + 3 \left(-1\right) + 1}\right|}{\sqrt{\left(-1\right)^{2} + 2^{2} + 3^{2}}} = \frac{9 \sqrt{14}}{14} \]
Substitute.✓ Proved

Answer \( \frac{9 \sqrt{14}}{14} \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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 foot of the perpendicular found numerically, then the distance to it

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and plane coefficients into the formula is accurate, and the algebraic simplification leads to the correct result.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies the point-to-plane distance formula. The substitution of coordinates and plane coefficients into the formula is accurate, and the algebraic simplification leads to the correct result.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The formula in step 1 is inconsistent with the substitution in step 2. Step 1 defines the numerator as |ax₀ + by₀ + cz₀ − d|, but step 2 computes |ax₀ + by₀ + cz₀ + d| (adding 1 instead of subtracting -1, or effectively treating the plane equation as equal to 1 rather than -1). While the final numerical answer happens to be correct because |-9| = |9|, the method described is algebraically wrong for the given plane equation -x + 3y + 2z = -1 (where d = -1, so one should subtract -1, i.e., add 1, but the formula says subtract d). Wait, let's look closer. Plane: -x + 3y + 2z = -1. Standard form ax+by+cz+d=0 is -x+3y+2z+1=0. Distance is |ax0+by0+cz0+d|/||n||. The formula in step 1 says |ax0+by0+cz0 - d|. This implies the plane is ax+by+cz=d. If plane is -x+3y+2z=-1, then d=-1. Formula says subtract d, so -(-1) = +1. Step 2 adds 1. So the substitution is actually consistent with the formula IF the formula is interpreted as 'subtract the RHS constant'. However, standard notation usually defines d in ax+by+cz+d=0. If the formula is |ax0+by0+cz0 - d|, it assumes the plane is ax+by+cz=d. Here -x+3y+2z=-1, so d=-1. Subtracting d means -(-1) = +1. Step 2 adds 1. So the calculation is correct. BUT, the formula |ax0+by0+cz0 - d| is ambiguous without defining d. Usually textbooks use |ax0+by0+cz0 + d| for ax+by+cz+d=0. If the student uses the formula from step 1, they must identify d as the RHS. Step 2 substitutes correctly for that interpretation. Is it misleading? It's a bit sloppy but not strictly wrong if d is defined as the RHS. Let's re-read carefully. Step 1: Distance = |ax₀ + by₀ + cz₀ − d| / ‖n‖. This is the standard formula for plane ax+by+cz=d. Step 2: Substitutes a=-1, b=3, c=2, x0=3, y0=-1, z0=-2. The term inside abs is (2*(-2) - 1*3 + 3*(-1)) + 1. Wait. The coefficients in step 2 are mixed up? Plane: -x + 3y + 2z. a=-1, b=3, c=2. Step 2 calculation: 2*(-2) is c*z0? No, c=2, z0=-2, so 2*-2=-4. -1*3 is a*x0? a=-1, x0=3, so -3. 3*(-1) is b*y0? b=3, y0=-1, so -3. Sum = -4-3-3 = -10. Then +1 = -9. Abs is 9. Denominator sqrt(1+4+9)=sqrt(14). Result 9/sqrt(14). This is correct. The only issue is the ordering of terms in the text of step 2 vs the variables. It writes 2*(-2) first (c*z), then -1*3 (a*x), then 3*(-1) (b*y). This is just a permutation, sum is commutative. The logic holds. Why did I think it was wrong? Ah, I see. Step 1 says |ax0+by0+cz0 - d|. Step 2 calculates |... + 1|. If d=-1, then -d = +1. So it is consistent. Verdict: pass.
gpt-oss:20b: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution substitutes the wrong coefficients into the distance formula. The plane is \(-x+3y+2z=-1\), so the normal vector is \((-1,3,2)\), not \((2,-1
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The formula in line 1 is ambiguous regarding the sign of d, and the substitution in line 2 is inconsistent with the standard form ax+by+cz=d. Specifically, if the plane is -x+3y+2z=-1, then d=-1, so the numerator should be |-1(3)+3(-1)+2(-2) - (-1)| = |-3-3-4+1| = 9. The solution computes (2*(-2) - 1*3 + 3*(-1)) + 1, which corresponds to evaluating the LHS and adding 1, effectively treating the equation as -x+3y+2z+1=0 or similar, but the coefficients in the numerator (2, -1, 3) do not match the normal vector (-1, 3, 2) used in the denominator's magnitude calculation (sqrt(1+4+9)). The numerator coefficients are swapped/incorrect relative to the denominator's normal vector components.
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 2026-10-18 with SymPy 1.14.0.