---
title: "Successive 2D rotations are additive"
subject: "Computer Graphics"
chapter: 3
key: "semester_5/computer_graphics:two-rotations-additive"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/two-rotations-additive
site: BCT Engineering
---

# Successive 2D rotations are additive

Computer Graphics · Chapter 3: Two –Dimensional Transformations

- Asked in: 2066 Magh (old course)
- Video: [Successive rotation || Computer Graphic || Bsc IT, BCA, MCA, BTech](https://www.youtube.com/watch?v=lh8ZLGTbEc8) — Techbuddy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/two-rotations-additive
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-3

## Question

Justify with necessary matrix operations that two successive rotations in 2-D is additive.

## Short answer

Multiply matrices: R(θ1)·R(θ2) yields entries cosθ1cosθ2−sinθ1sinθ2 and sinθ1cosθ2+cosθ1sinθ2, which equal cos(θ1+θ2) and sin(θ1+θ2). Hence R(θ1)·R(θ2)=R(θ1+θ2), proving additivity.

## Answer

**Claim:** R(θ1) · R(θ2) = R(θ1 + θ2), so two successive rotations by θ1 and θ2 are the same as one rotation by θ1 + θ2.

2D rotation about the origin (homogeneous form):

```text
R(θ) = | cosθ  -sinθ  0 |
       | sinθ   cosθ  0 |
       | 0      0     1 |
```

Rotating first by θ2 and then by θ1 gives the product R(θ1)·R(θ2):

```text
R(θ1)·R(θ2) =
| cosθ1cosθ2 - sinθ1sinθ2   -(cosθ1sinθ2 + sinθ1cosθ2)   0 |
| sinθ1cosθ2 + cosθ1sinθ2    cosθ1cosθ2 - sinθ1sinθ2    0 |
| 0                          0                          1 |
```

Using the trigonometric identities
- cos(θ1 + θ2) = cosθ1cosθ2 − sinθ1sinθ2
- sin(θ1 + θ2) = sinθ1cosθ2 + cosθ1sinθ2

```text
R(θ1)·R(θ2) = | cos(θ1+θ2)  -sin(θ1+θ2)  0 |
              | sin(θ1+θ2)   cos(θ1+θ2)  0 | = R(θ1 + θ2)
              | 0            0           1 |
```

Hence successive rotations are additive. Since θ1 + θ2 = θ2 + θ1, rotations about the same pivot point are also commutative.

## Check yourself

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

1. What is being shown?
   A. R(θ1)·R(θ2) = R(θ1·θ2)
   B. Rotation equals scaling
   C. R is its own inverse
   D. R(θ1)·R(θ2) = R(θ1+θ2)
   Answer: D. Successive rotations add their angles. (Key point: Claim)

2. Which identities are used?
   A. cos and sin of sum of angles
   B. Pythagorean identity only
   C. Double angle only
   D. tan identities only
   Answer: A. cos(θ1+θ2) and sin(θ1+θ2) appear in the product. (Key point: Tools)

3. The top-left entry of the product is?
   A. cosθ1cosθ2 + sinθ1sinθ2
   B. cosθ1cosθ2 - sinθ1sinθ2
   C. sinθ1cosθ2
   D. cosθ1 + cosθ2
   Answer: B. It equals cos(θ1+θ2). (Key point: Top-left entry)

4. Since θ1 + θ2 = θ2 + θ1, rotations about the same point are?
   A. Non-commutative
   B. Inverse
   C. Commutative
   D. Singular
   Answer: C. Order does not matter. (Key point: Consequence)

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