---
title: "Angle between normals to surfaces"
subject: "Mathematics"
chapter: 5
key: "semester_2/mathematics:angle-between-normals-surfaces"
source: https://www.bctengineer.com/q/semester_2/mathematics/angle-between-normals-surfaces
site: BCT Engineering
---

# Angle between normals to surfaces

Mathematics · Chapter 5: Vector Algebra and Calculus

- Asked in: 2075 Baisakh, 5 marks; 2069 Bhadra, 5 marks (asked 2 times)
- Video: [Vector calculus find the angle between the normal's to the surfaces](https://www.youtube.com/watch?v=GA3mck8zuJU) — EASY MATHS EASY TRICKS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/angle-between-normals-surfaces
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-5

## Question

Find the angle between the normal to the surfaces given by: x log z = y² − 1 and x²y + z = 2 at the point (1,1,1).

## Short answer

Normals: n₁ = ∇(x log z - y² +1) = (0, -2, 1), n₂ = ∇(x²y + z -2) = (2, 1, 1). cos θ = |n₁·n₂|/(|n₁||n₂|) = 1/√30, so acute angle between normals is cos⁻¹(1/√30) ≈ 79.5°.

## Answer

The normal to a surface f(x, y, z) = 0 at a point is along ∇f. The angle between two surfaces at a point is the angle between their normals there, found from cos θ = (n₁ · n₂)/(|n₁||n₂|).

#### Surface 1: f = x log z − y² + 1 = 0
∇f = (log z, −2y, x/z).
At (1, 1, 1): log 1 = 0, so n₁ = (0, −2, 1). (The point lies on the surface: 0 − 1 + 1 = 0 ✓.)

#### Surface 2: g = x²y + z − 2 = 0
∇g = (2xy, x², 1).
At (1, 1, 1): n₂ = (2, 1, 1). (1 + 1 − 2 = 0 ✓.)

#### Angle
```text
n₁ · n₂ = 0·2 + (−2)(1) + (1)(1) = −1
|n₁| = √5,   |n₂| = √6
cos θ = −1/√30
```
θ = cos⁻¹(−1/√30) ≈ 100.5° between these two normal directions. The acute angle between the surfaces (taking the normals in the other sense) is cos⁻¹(1/√30) ≈ 79.5°.

**Answer: the angle between the normals is cos⁻¹(1/√30) ≈ 79.5° (acute angle; the obtuse value from the signs of the chosen normals is cos⁻¹(−1/√30) ≈ 100.5°).**

## Check yourself

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

1. The angle between two surfaces at a point is:
   A. the angle between ∇f and the position vector
   B. the angle between tangent lines only
   C. the angle between their normals ∇f and ∇g
   D. always 90°
   Answer: C. Normals are along gradients. (Key point: Method)

2. For x log z − y^2 + 1 = 0, n1 at (1,1,1) is:
   A. (0, 2, 1)
   B. (0, −2, 1)
   C. (1, −2, 0)
   D. (log z, −2, 1)
   Answer: B. ∇f = (log z, −2y, x/z) and log 1 = 0. (Key point: Normal 1)

3. n1 · n2 for n1 = (0,−2,1), n2 = (2,1,1) is:
   A. −1
   B. 1
   C. −3
   D. 0
   Answer: A. 0 − 2 + 1 = −1. (Key point: Dot product)

4. The acute angle between the surfaces is:
   A. cos⁻¹(1/√6) ≈ 66°
   B. cos⁻¹(−1/√30) only
   C. 90°
   D. cos⁻¹(1/√30) ≈ 79.5°
   Answer: D. |n1| = √5, |n2| = √6, so cos θ = 1/√30 for the acute angle. (Key point: Answer)

## Related in Mathematics

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