---
title: "Angle between surfaces x²+y²+z²=9 and z=x²+y²−3"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:angle-between-surfaces-x2-y2"
source: https://www.bctengineer.com/q/semester_2/mathematics/angle-between-surfaces-x2-y2
site: BCT Engineering
---

# Angle between surfaces x²+y²+z²=9 and z=x²+y²−3

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2069 Poush, 5 marks
- Video: [x² + y² + z² = 9 and z = x² + y² − 3 | Angle Between Surfaces Problem](https://www.youtube.com/watch?v=asKR1YY_PpQ) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/angle-between-surfaces-x2-y2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Find the angle between the surfaces x²+y²+z² = 9 and z = x²+y²−3 at the point (2,−1,2).

## Short answer

Normals from gradients: sphere ∇f = (4,−2,4), paraboloid ∇g = (4,−2,−1). cosθ = (16+4−4)/(6√21) = 16/(6√21) = 8/(3√21). The angle is θ = cos⁻¹(8/(3√21)) ≈ 54.4°.

## Answer

The angle between two surfaces at a common point is the angle between their normals there. For f(x, y, z) = 0 the normal is ∇f.

#### Check the point
- Sphere: 4 + 1 + 4 = 9 ✓.
- Paraboloid z = x² + y² − 3: 4 + 1 − 3 = 2 = z ✓.

#### Normals
Surface 1: f = x² + y² + z² − 9, ∇f = (2x, 2y, 2z). At (2, −1, 2): n₁ = (4, −2, 4).

Surface 2: g = x² + y² − z − 3, ∇g = (2x, 2y, −1). At (2, −1, 2): n₂ = (4, −2, −1).

#### Angle
```text
n₁ · n₂ = 16 + 4 − 4 = 16
|n₁| = √(16 + 4 + 16) = 6
|n₂| = √(16 + 4 + 1) = √21
cos θ = 16 / (6√21) = 8 / (3√21) = 8√21 / 63 ≈ 0.5819
θ ≈ 54.4°
```

**Answer: θ = cos⁻¹(8/(3√21)) ≈ 54.4°.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. At (2,−1,2), n1 for the sphere is:
   A. (2, −1, 2)
   B. (4, −2, 4)
   C. (4, −2, −1)
   D. (4, 2, 4)
   Answer: B. ∇f = (2x, 2y, 2z). (Key point: Normals)

2. n2 for z = x²+y²−3 is:
   A. (4, −2, −1)
   B. (4, −2, 1)
   C. (4, −2, 4)
   D. (2, −1, 0)
   Answer: A. g = x²+y²−z−3 gives (2x, 2y, −1). (Key point: Second normal)

3. n1 · n2 is:
   A. 24
   B. 8
   C. 36
   D. 16
   Answer: D. 16 + 4 − 4. (Key point: Dot product)

4. The angle is:
   A. cos⁻¹(16/21) ≈ 40°
   B. 90°
   C. cos⁻¹(8/(3√21)) ≈ 54.4°
   D. cos⁻¹(1/2) = 60°
   Answer: C. |n1| = 6, |n2| = √21. (Key point: Angle)

## Related in Mathematics

- [Angle between normals to surfaces](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aangle-between-normals-surfaces) — Mathematics, chapter 5, asked 2×

## Related in other subjects

- [Critical clearing angle, breakers 1 and 2](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Acritical-clearing-angle-breakers-1-2) — Power System Analysis II, chapter 6, asked 2×
- [FBD; tensions in strings AC, CD, BD and angle θ](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Afree-body-diagram-strings-ac-cd-bd-theta) — Applied Mechanics, chapter 2
- [Critical clearing time and angle, fault at P](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Acritical-clearing-time-point-p-h5) — Power System Analysis II, chapter 6
- [Angle criterion for root locus](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-angle-criterion) — Control Systems, chapter 5