Planes: distances and angles
Problem 9.769 · medium
Find the angle between the planes \( \displaystyle x + 4 y + 4 z = 0 \) and \( \displaystyle x + 3 y + 2 z = 0 \).
- The angle between planes is the (acute) angle between their normals.Reviewed
- \[ \frac{\left|{1 \cdot 1 + 4 \cdot 2 + 4 \cdot 3}\right|}{\sqrt{1^{2} + 2^{2} + 3^{2}} \sqrt{1^{2} + 4^{2} + 4^{2}}} = \frac{\sqrt{462}}{22} \]cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
- θ = arccos of that: acos(sqrt(462)/22).Reviewed
Answer \( \operatorname{acos}{\left(\frac{\sqrt{462}}{22} \right)} \approx 12.31^\circ \)
Lines: 1 proved, 2 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 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the normals' angle in floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the angle between planes as the angle between their normal vectors and applies the standard formula involving the dot product and magnitudes. The use of the absolute value ensures the acute angle is found, consistent with the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the angle between planes as the angle between their normal vectors and applies the standard formula involving the dot product and magnitudes. The use of the absolute value ensures the acute angle is found, consistent with the stated answer.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The dot product calculation in line 2 is incorrect; it computes 1*1 + 4*2 + 4*3 = 19, but the correct dot product of the normal vectors (1, 4, 4) and (1, 3, 2) is 1*1 + 4*3 + 4*2 = 19. Wait, 1*1 + 4*3 + 4*2 = 1 + 12 + 8 = 21. The solution used 4*2 and 4*3 instead of 4*3 and 4*2? No, it used 4*2 (for y) and 4*3 (for z). The normal for the second plane is (1, 3, 2). The normal for the first is (1, 4, 4). The dot product is 1*1 + 4*3 + 4*2 = 1 + 12 + 8 = 21. The solution calculated 1*1 + 4*2 + 4*3 = 1 + 8 + 12 = 21. Actually, the arithmetic in the equation line `1*1 + 4*2 + 4*3` sums to 21. Let's re-read carefully. The equation says `Abs(1*1 + 4*2 + 4*3)`. This is `1 + 8 + 12 = 21`. The denominator is `sqrt(1+4+9)*sqrt(1+16+16) = sqrt(14)*sqrt(37) = sqrt(518)`. The solution claims the result is `sqrt(462)/22`. `sqrt(462)/22` squared is `462/484 = 231/242`. `21/sqrt(518)` squared is `441/518 = 63/74`. These are not equal. The solution's equation is mathematically false. The dot product terms are swapped relative to the vector components if we assume standard order, but more importantly, the final value `sqrt(462)/22` is incorrect for these planes. The correct cosine is `21/sqrt(518)`. The provided answer `sqrt(462)/22` corresponds to a different problem or calculation error. Specifically, `sqrt(462)/22` simplifies to `sqrt(462)/22`. Let's check if `21/sqrt(518)` equals `sqrt(462)/22`. `21^2 = 441`. `518`. `462/22^2 = 462/484`. `441/518 != 462/484`. So the equation is false.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The dot product calculation in line 2 is incorrect; it computes 1*1 + 4*2 + 4*3 = 19, but the correct dot product of the normal vectors <1, 4, 4> and <1, 3, 2> is 1*1 + 4*3 + 4*2 = 19. Wait, 1+12+8=21. The solution uses 19. Let me re-read carefully. Line 2 says `1*1 + 4*2 + 4*3`. The normals are n1=<1,4,4> and n2=<1,3,2>. The dot product is 1*1 + 4*3 + 4*2 = 1 + 12 + 8 = 21. The solution calculates 1*1 + 4*2 + 4*3 = 1 + 8 + 12 = 21. Ah, the order of terms in the sum is swapped but the values are correct? No. n1 components are 1, 4, 4. n2 components are 1, 3, 2. Dot product is 1*1 + 4*3 + 4*2. The solution writes `1*1 + 4*2 + 4*3`. This corresponds to pairing n1_y with n2_z and n1_z with n2_y. 4*2=8 and 4*3=12. Sum is 21. So the arithmetic result 21 is correct, but the pairing is wrong? Actually, 4*2 + 4*3 = 8+12=20. 1+20=21. So the numerical value is correct. However, the denominator calculation in the text `sqrt(1**2 + 2**2 + 3**2)` implies the second normal is <1,2,3>, but the plane is x+3y+2z=0, so the normal is <1,3,2>. The norm squared is 1+9+4=14. The solution writes `2**2 + 3**2` which is 4+9=13. 1+13=14. So the numerical value of the denominator is correct (sqrt(14)), but the components listed are wrong (2 and 3 instead of 3 and 2). Since 2^2+3^2 = 3^2+2^2, the magnitude is correct. The dot product terms 4*2 and 4*3 also sum to the same value as 4*3 and 4*2. So the final number is correct. Is it an error? It's a sloppy derivation that happens to yield the right number due to symmetry of squares and commutativity of addition. But wait, does it teach something false? It implies the normal vector is <1,2,3> or that the dot product pairs are mixed up. It is misleading because it misidentifies the components of the normal vector in the formula application, even if the final arithmetic coincidentally matches.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.