---
title: "3D rotation matrices (clockwise)"
subject: "Computer Graphics"
chapter: 4
key: "semester_5/computer_graphics:rotation-matrices-xyz-clockwise"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/rotation-matrices-xyz-clockwise
site: BCT Engineering
---

# 3D rotation matrices (clockwise)

Computer Graphics · Chapter 4: Three-Dimensional Graphics

- Asked in: 2071 Chaitra, 3 marks
- Video: [3D Rotation | CG | Computer Graphics | Lec-28 | Bhanu Priya](https://www.youtube.com/watch?v=MjUt6f_OWXg) — Education 4u
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/rotation-matrices-xyz-clockwise
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-4

## Question

Write rotation matrix in clockwise direction with respect to x-axis, y-axis and z-axis.

## Short answer

Clockwise rotation matrices use −θ. x-axis: |1 0 0 0; 0 cosθ sinθ 0; 0 −sinθ cosθ 0; 0 0 0 1|; y-axis: |cosθ 0 −sinθ 0; 0 1 0 0; sinθ 0 cosθ 0; 0 0 0 1|; z-axis: |cosθ sinθ 0 0; −sinθ cosθ 0 0; 0 0 1 0; 0 0 0 1|.

## Answer

Standard (anti-clockwise) rotation matrices use positive θ. For clockwise rotation replace θ by −θ, using cos(−θ) = cosθ and sin(−θ) = −sinθ.

**About the x-axis (clockwise)**

```text
Rx(−θ) = | 1  0      0     0 |
         | 0  cosθ   sinθ  0 |
         | 0 -sinθ   cosθ  0 |
         | 0  0      0     1 |
```
y' = y cosθ + z sinθ, z' = −y sinθ + z cosθ, x' = x

**About the y-axis (clockwise)**

```text
Ry(−θ) = | cosθ  0 -sinθ  0 |
         | 0     1  0     0 |
         | sinθ  0  cosθ  0 |
         | 0     0  0     1 |
```
x' = x cosθ − z sinθ, z' = x sinθ + z cosθ, y' = y

**About the z-axis (clockwise)**

```text
Rz(−θ) = |  cosθ  sinθ  0  0 |
         | -sinθ  cosθ  0  0 |
         |  0     0     1  0 |
         |  0     0     0  1 |
```
x' = x cosθ + y sinθ, y' = −x sinθ + y cosθ, z' = z

Each is the transpose (and inverse) of the matching anticlockwise matrix.

## Check yourself

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

1. How is a clockwise rotation matrix obtained from the standard one?
   A. Replace θ by -θ
   B. Swap rows and columns of the identity
   C. Multiply by -1
   D. Replace cosθ by sinθ
   Answer: A. Using cos(-θ) = cosθ and sin(-θ) = -sinθ flips the sign of the sine terms. (Key point: Clockwise rule)

2. Clockwise rotation about z has first row:
   A. cosθ, -sinθ, 0, 0
   B. cosθ, sinθ, 0, 0
   C. -sinθ, cosθ, 0, 0
   D. sinθ, cosθ, 0, 0
   Answer: B. Rz(-θ) has x' = x cosθ + y sinθ. (Key point: Clockwise Rz)

3. The clockwise matrix is related to the anticlockwise one as:
   A. Its negative
   B. Its square
   C. Its transpose and inverse
   D. Its determinant-scaled copy
   Answer: C. Rotation matrices are orthogonal, and reversing the angle gives the inverse = transpose. (Key point: Relation)

## Related in Computer Graphics

- [3D rotation about axis (1,2,3)-(10,20,30)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotate-3d-axis-1-2-3) — Computer Graphics, chapter 4
- [3D rotation about arbitrary axis](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotate-3d-arbitrary-axis-steps) — Computer Graphics, chapter 4, asked 2×
- [3D rotation steps and perspective projection](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotate-3d-arbitrary-axis-perspective) — Computer Graphics, chapter 4
- [3D shearing and matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Awhat-is-3d-shearing) — Computer Graphics, chapter 4, asked 3×
- [3D world to viewing coordinate matrix](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aworld-to-viewing-3d-matrix) — Computer Graphics, chapter 4, asked 2×
- [Rotation about a pivot point](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotation-about-pivot-point-derive) — Computer Graphics, chapter 3, asked 2×

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- [Anti-clockwise traverse PQRSP missing data](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanticlockwise-traverse-pqrsp-missing) — Surveying II, chapter 1