Home›Calculus 3›Planes: distances and angles›Problem 9.721

Planes: distances and angles

Problem 9.721 · medium

Find the angle between the planes \( \displaystyle - 4 x + 2 y - 4 z = -5 \) and \( \displaystyle x - 2 y + 2 z = -1 \).

The angle between planes is the (acute) angle between their normals.
\[ \frac{\left|{\left(-4\right) 2 - 4 \cdot 1 + 2 \left(-2\right)}\right|}{\sqrt{1^{2} + \left(-2\right)^{2} + 2^{2}} \sqrt{2^{2} + \left(-4\right)^{2} + \left(-4\right)^{2}}} = \frac{8}{9} \]
cos θ = |n₁ · n₂| / (‖n₁‖‖n₂‖).✓ Proved
θ = arccos of that: acos(8/9).

Answer \( \operatorname{acos}{\left(\frac{8}{9} \right)} \approx 27.27^\circ \)

Lines: 1 proved, 2 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
3	Not checked	—	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: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution computes the cosine of the angle between the normal vectors but fails to verify that the resulting angle is acute. While the absolute value in the dot product formula ensures the cosine is positive (and thus the angle is acute), the text explicitly states the angle is the acute angle between normals, but the calculation `Abs(...)` is applied to the dot product, not the final angle. More critically, the problem asks for the angle between planes, which is conventionally defined as the acute angle between their normal vectors. The calculation `8/9` is correct for `cos(theta)`. However, the solution does not explicitly state that the angle between planes is defined as the acute angle, relying on the absolute value trick. The main issue is that the solution is technically correct in its result, but the reasoning in step 1 is slightly imprecise: the angle between planes is the acute angle between normals, but if the normals form an obtuse angle, the angle between planes is the supplement. The absolute value handles this. The real error is subtle: the solution claims the angle is `acos(8/9)`. `8/9` is positive, so `acos(8/9)` is acute. This is correct. Wait, let me re-read the prompt's constraints. 'Find what is wrong'. Is there an error? Let's look at the normals: n1 = <-4, 2, -4>, n2 = <1, -2, 2>. n1 . n2 = -4 - 4 - 8 = -16. |n1| = sqrt(16+4+16) = sqrt(36) = 6. |n2| = sqrt(1+4+4) = sqrt(9) = 3. cos(theta) = -16 / 18 = -8/9. The angle between the normals is arccos(-8/9), which is obtuse. The angle between the planes is the acute angle, so we take the absolute value of the cosine, or the supplement. The acute angle theta satisfies cos(theta) = 8/9. So theta = arccos(8/9). The solution is correct. Is there a style issue? 'The angle between planes is the (acute) angle between their normals.' This is standard. Step 2 calculates the absolute value of the dot product divided by magnitudes. Step 3 takes arccos. The result is correct. Let me look closer at the equation line. `Abs(-4*2 - 4*1 + 2*(-2))`. Wait. The dot product is `n1 . n2`. n1 = <-4, 2, -4>. n2 = <1, -2, 2>. Dot product = (-4)(1) + (2)(-2) + (-4)(2) = -4 - 4 - 8 = -16. The equation in line 2 says: `Abs(-4*2 - 4*1 + 2*(-2))`. This looks like it's computing `Abs( n1_x * n2_y ? No. Let's trace the numbers. -4*2 = -8. (n1_x * n2_z? No, n2_z is 2. n1_x is -4. So -4*2 is n1_x * n2_z? Or n1_z * n2_x? n1_z is -4, n2_x is 1. No. Let's look at the terms: `-4*2`, `-4*1`, `2*(-2)`. If n1 = <-4, 2, -4> and n2 = <1, -2, 2>. Term 1: -4*2. This could be n1_x * n2_z (-4*2) or n1_z * n2_x (-4*1 is next). Term 2: -4*1. This is n1_z * n2_x? Or n1_x * n2_x? n1_x is -4, n2_x is 1. So -4*1 is n1_x * n2_x. Term 3: 2*(-2). This is n1_y * n2_y (2 * -2). So the expression inside Abs is `n1_x * n2_z + n1_z * n2_x + n1_y * n2_y`? No, that's not the dot product. The dot product is `n1_x*n2_x + n1_y*n2_y + n1_z*n2_z`. `n1_x*n2_x = -4*1 = -4`. `n1_y*n2_y = 2*-2 = -4`. `n1_z*n2_z = -4*2 = -8`. Sum = -16. The expression in the solution is: `-4*2 - 4*1 + 2*(-2)` = `-8 - 4 - 4 = -16`. Wait, `-4*2` is -8. `-4*1` is -4. `2*-2` is -4. Sum is -16. But the terms are mixed up. It calculated `n1_x * n2_z` (-4*2) instead of `n1_x * n2_x` (-4*1)? No, it has `-4*1` as the second term. It has `-4*2` as the first term. It has `2*-2` as the third term. Standard dot product: `(-4)(1) + (2)(-2) + (-4)(2) = -4 - 4 - 8 = -16`. The solution's expression: `(-4)(2) + (-4)(1) + (2)(-2) = -8 - 4 - 4 = -16`. It seems to have swapped the z-component of n1 with the x-component of n1 in the first term? Or swapped n2's components? Actually, it looks like it computed `n1 . n2` but permuted the indices incorrectly yet got the same numerical result by coincidence? Or did it just write the terms in a weird order? Let's check the terms again. Solution: `-4*2` (n1_x * n2_z?), `-4*1` (n1_z * n2_x?), `2*-2` (n1_y * n2_y). If it meant `n1_x * n2_x`, it should be `-4*1`. If it meant `n1_z * n2_z`, it should be `-4*2`. The solution has `-4*2` AND `-4*1`. It has two terms with -4. n1 has two -4s. n2 has one 1 and one 2. So it paired n1_x(-4) with n2_z(2) -> -8. It paired n1_z(-4) with n2_x(1) -> -4. It paired n1_y(2) with n2_y(-2) -> -4. Sum = -16. This is NOT the dot product. The dot product is `n1_x n2_x + n1_y n2_y + n1_z n2_z`. `n1_x n2_x = -4*1 = -4`. `n1_z n2_z = -4*2 = -8`. The solution calculated `n1_x n2_z + n1_z n2_x + n1_y n2_y`. This is a 'cross' term sum, not the dot product. It happens to yield the same magnitude because the values are symmetric in a way? No, -4 and -8 vs -8 and -4. Addition is commutative. So `-4 + -8 = -12`. `-8 + -4 = -12`. The sum is the same. So the numerical value is correct, but the formula applied is wrong (it's not the dot product formula, it's a scrambled version that accidentally works because the non-matching components were swapped and the values happened to make the sum identical? No, `a*b + c*d` vs `a*d + c*b`. Here `a=-4, b=1, c=-4, d=2`. `a*b+c*d = -4-8=-12`. `a*d+c*b = -8-4=-12`. Yes, it works by coincidence because the two components of n1 were equal (-4) and the two components of n2 were distinct (1, 2). If n1 was <-4, 2, -5>, this would fail. So the sentence/equation line applies the wrong formula (scrambled dot product) even though the number is right. This is a 'misleading' or 'error' depending on severity. It teaches a student that you can multiply x with z and z with x and get the dot product. That is false. Therefore, it is an error in the method shown.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution computes the cosine of the angle between the normal vectors but fails to verify that the resulting angle is acute. While the absolute value in the dot product formula ensures the cosine is positive (and thus the angle is acute), the text explicitly states the angle is the acute angle between normals, but the calculation `Abs(...)` is applied to the dot product, not the final angle. More critically, the problem asks for the angle between planes, which is conventionally defined as the acute angle between their normal vectors. The calculation `8/9` is correct for `cos(theta)`. However, the solution does not explicitly state that the angle between planes is defined as the acute angle, relying on the absolute value trick. The main issue is that the solution is technically correct in its result, but the reasoning in step 1 is slightly imprecise: the angle between planes is the acute angle between normals, but if the normals form an obtuse angle, the angle between planes is the supplement. The absolute value handles this. The real error is subtle: the solution claims the angle is `acos(8/9)`. `8/9` is positive, so `acos(8/9)` is acute. This is correct. Wait, let me re-read the prompt's constraints. 'Find what is wrong'. Is there an error? Let's look at the normals: n1 = <-4, 2, -4>, n2 = <1, -2, 2>. n1 . n2 = -4 - 4 - 8 = -16. |n1| = sqrt(16+4+16) = sqrt(36) = 6. |n2| = sqrt(1+4+4) = sqrt(9) = 3. cos(theta) = -16 / 18 = -8/9. The angle between the normals is arccos(-8/9), which is obtuse. The angle between the planes is the acute angle, so we take the absolute value of the cosine, or the supplement. The acute angle theta satisfies cos(theta) = 8/9. So theta = arccos(8/9). The solution is correct. Is there a style issue? 'The angle between planes is the (acute) angle between their normals.' This is standard. Step 2 calculates the absolute value of the dot product divided by magnitudes. Step 3 takes arccos. The result is correct. Let me look closer at the equation line. `Abs(-4*2 - 4*1 + 2*(-2))`. Wait. The dot product is `n1 . n2`. n1 = <-4, 2, -4>. n2 = <1, -2, 2>. Dot product = (-4)(1) + (2)(-2) + (-4)(2) = -4 - 4 - 8 = -16. The equation in line 2 says: `Abs(-4*2 - 4*1 + 2*(-2))`. This looks like it's computing `Abs( n1_x * n2_y ? No. Let's trace the numbers. -4*2 = -8. (n1_x * n2_z? No, n2_z is 2. n1_x is -4. So -4*2 is n1_x * n2_z? Or n1_z * n2_x? n1_z is -4, n2_x is 1. No. Let's look at the terms: `-4*2`, `-4*1`, `2*(-2)`. If n1 = <-4, 2, -4> and n2 = <1, -2, 2>. Term 1: -4*2. This could be n1_x * n2_z (-4*2) or n1_z * n2_x (-4*1 is next). Term 2: -4*1. This is n1_z * n2_x? Or n1_x * n2_x? n1_x is -4, n2_x is 1. So -4*1 is n1_x * n2_x. Term 3: 2*(-2). This is n1_y * n2_y (2 * -2). So the expression inside Abs is `n1_x * n2_z + n1_z * n2_x + n1_y * n2_y`? No, that's not the dot product. The dot product is `n1_x*n2_x + n1_y*n2_y + n1_z*n2_z`. `n1_x*n2_x = -4*1 = -4`. `n1_y*n2_y = 2*-2 = -4`. `n1_z*n2_z = -4*2 = -8`. Sum = -16. The expression in the solution is: `-4*2 - 4*1 + 2*(-2)` = `-8 - 4 - 4 = -16`. Wait, `-4*2` is -8. `-4*1` is -4. `2*-2` is -4. Sum is -16. But the terms are mixed up. It calculated `n1_x * n2_z` (-4*2) instead of `n1_x * n2_x` (-4*1)? No, it has `-4*1` as the second term. It has `-4*2` as the first term. It has `2*-2` as the third term. Standard dot product: `(-4)(1) + (2)(-2) + (-4)(2) = -4 - 4 - 8 = -16`. The solution's expression: `(-4)(2) + (-4)(1) + (2)(-2) = -8 - 4 - 4 = -16`. It seems to have swapped the z-component of n1 with the x-component of n1 in the first term? Or swapped n2's components? Actually, it looks like it computed `n1 . n2` but permuted the indices incorrectly yet got the same numerical result by coincidence? Or did it just write the terms in a weird order? Let's check the terms again. Solution: `-4*2` (n1_x * n2_z?), `-4*1` (n1_z * n2_x?), `2*-2` (n1_y * n2_y). If it meant `n1_x * n2_x`, it should be `-4*1`. If it meant `n1_z * n2_z`, it should be `-4*2`. The solution has `-4*2` AND `-4*1`. It has two terms with -4. n1 has two -4s. n2 has one 1 and one 2. So it paired n1_x(-4) with n2_z(2) -> -8. It paired n1_z(-4) with n2_x(1) -> -4. It paired n1_y(2) with n2_y(-2) -> -4. Sum = -16. This is NOT the dot product. The dot product is `n1_x n2_x + n1_y n2_y + n1_z n2_z`. `n1_x n2_x = -4*1 = -4`. `n1_z n2_z = -4*2 = -8`. The solution calculated `n1_x n2_z + n1_z n2_x + n1_y n2_y`. This is a 'cross' term sum, not the dot product. It happens to yield the same magnitude because the values are symmetric in a way? No, -4 and -8 vs -8 and -4. Addition is commutative. So `-4 + -8 = -12`. `-8 + -4 = -12`. The sum is the same. So the numerical value is correct, but the formula applied is wrong (it's not the dot product formula, it's a scrambled version that accidentally works because the non-matching components were swapped and the values happened to make the sum identical? No, `a*b + c*d` vs `a*d + c*b`. Here `a=-4, b=1, c=-4, d=2`. `a*b+c*d = -4-8=-12`. `a*d+c*b = -8-4=-12`. Yes, it works by coincidence because the two components of n1 were equal (-4) and the two components of n2 were distinct (1, 2). If n1 was <-4, 2, -5>, this would fail. So the sentence/equation line applies the wrong formula (scrambled dot product) even though the number is right. This is a 'misleading' or 'error' depending on severity. It teaches a student that you can multiply x with z and z with x and get the dot product. That is false. Therefore, it is an error in the method shown.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution computes the cosine of the angle between the normal vectors, but fails to verify that the resulting angle is acute. While the absolute
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution computes the cosine of the angle between the normal vectors, but the angle between the planes is defined as the acute angle between them. Since the dot product of the normals is negative (-8), the angle between normals is obtuse. The formula used in step 2 correctly takes the absolute value to find the acute angle, but step 1's statement is slightly imprecise without mentioning the acute angle constraint explicitly in the context of the calculation, though the calculation itself is correct. However, looking closer at step 2, the equation is marked 'unchecked' but the prompt says equations are checked by a CAS. The real issue is that the problem asks for 'the angle', which is conventionally the acute one. The calculation `Abs(...)` yields 8/9. `acos(8/9)` is indeed the acute angle. Wait, let's re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. Step 2 is an equation. Step 1 and 3 are sentences. Step 1 says 'The angle between planes is the (acute) angle between their normals.' This is correct. Step 3 says 'theta = arccos of that'. This is correct. Is there an error? Let's check the dot product. n1 = <-4, 2, -4>, n2 = <1, -2, 2>. n1.n2 = -4 -4 -8 = -16. |n1| = sqrt(16+4+16) = sqrt(36) = 6. |n2| = sqrt(1+4+4) = 3. cos(theta_normals) = -16/18 = -8/9. The angle between normals is arccos(-8/9). The angle between planes is the acute angle, so we take the absolute value of the cosine, 8/9. The solution does this. Why would it fail? Ah, look at step 2's equation text: `Abs(-4*2 - 4*1 + 2*(-2))`. The dot product is `(-4)(1) + (2)(-2) + (-4)(2) = -4 -4 -8 = -16`. The equation in step 2 writes `-4*2 - 4*1 + 2*(-2)`. This corresponds to components: x: -4*2? No, n1_x is -4, n2_x is 1. The term is -4*1. The equation has -4*2. It seems to have swapped components or used wrong values. Let's look at the numbers in the equation: `-4*2` (maybe n1_x * n2_y?), `-4*1` (n1_z * n2_x?), `2*(-2)` (n1_y * n2_y?). The dot product calculation in the equation is garbled/wrong terms, even if the result 8/9 is coincidentally correct or derived from a different error? Let's re-evaluate the equation's RHS. `sqrt(1**2 + (-2)**2 + 2**2)` is |n2|=3. `sqrt(2**2 + (-4)**2 + (-4)**2)` is sqrt(4+16+16)=6. Denom is 18. Numerator in equation: `-8 -4 -4 = -16`. Abs is 16. 16/18 = 8/9. So the arithmetic in the equation `Abs(-4*2 - 4*1 + 2*(-2))` evaluates to `Abs(-8 -4 -4) = 16`. This matches the correct numerator magnitude. BUT, the terms `-4*2`, `-4*1`, `2*(-2)` do not correspond to the dot product of `<-4, 2, -4>` and `<1, -2, 2>`. The dot product is `(-4)(1) + (2)(-2) + (-4)(2)`. The equation uses `(-4)(2) + (-4)(1) + (2)(-2)`. It seems to have used n1 = <-4, -4, 2> or something? No, it just mixed up the multiplication pairs. It multiplied n1_x by n2_z? (-4*2). n1_z by n2_x? (-4*1). n1_y by n2_y? (2*-2). This is not the dot product formula. It is a computational error in the setup of the equation, even if the final number is right by coincidence (since multiplication is commutative, the set of products is the same: {-4, 2, -4} x {1, -2, 2} -> pairs are (-4,1), (2,-2), (-4,2). The equation uses (-4,2), (-4,1), (2,-2). The set of factors is the same, so the sum is the same. So the algebra is technically correct in result, but the representation of the dot product is nonsensical/wrongly ordered. However, usually 'unchecked' means we don't check the algebra. But the prompt says 'equation... checked by CAS... do not re-check the algebra'. So I should assume the equation is true. If the equation is true, and the sentences are correct, it passes. Is there a subtle error? 'The angle between planes is the (acute) angle between their normals.' This is standard. 'theta = arccos of that'. Standard. The answer is acos(8/9). This is correct. Why would it be a fail? Maybe the prompt implies I should check if the sentences are logically sound. They are. Maybe the 'unchecked' tag is a distractor. Let's look for 'misleading'. Does it teach something false? No. Is it 'style'? No. I will mark it pass.
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.