---
title: "Rotate object about axis y=4"
subject: "Computer Graphics"
chapter: 4
key: "semester_5/computer_graphics:rotate-object-about-y-4"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/rotate-object-about-y-4
site: BCT Engineering
---

# Rotate object about axis y=4

Computer Graphics · Chapter 4: Three-Dimensional Graphics

- Asked in: 2071 Chaitra, 7 marks
- Video: [3D Rotation about an axis parallel to a coordinate axis](https://www.youtube.com/watch?v=u7JzCdNpKTk) — sharvali sarnaik
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/rotate-object-about-y-4
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-4

## Question

Rotate the object (0, 0, 0), (2, 3, 0), (5, 0, 4) about the rotation axis y = 4.

## Short answer

Translate axis to x-axis, rotate 90° about x, translate back. For θ=90°, points become (0,4,-4), (2,4,-1), (5,0,-4).

## Answer

**Assumptions:** the rotation axis is the line y = 4 (z = 0), which is parallel to the x-axis. The angle is not given, so θ = 90° anticlockwise is used (the general matrix is given as well).

**Method:** translate the axis to the x-axis, rotate about x, translate back.

M = T(0, 4, 0) · Rx(θ) · T(0, −4, 0)

```text
         | 1 0 0  0 |
Rx(θ) =  | 0 c -s 0 |      c = cosθ, s = sinθ
         | 0 s  c 0 |
         | 0 0 0  1 |
```

General result: y' = 4 + (y − 4)cosθ − z sinθ, z' = (y − 4)sinθ + z cosθ, x' = x.

For θ = 90° (c = 0, s = 1): y' = 4 − z, z' = y − 4.

```text
M = | 1 0 0 0 |
    | 0 0 -1 4 |
    | 0 1 0 -4 |
    | 0 0 0 1 |
```

| Point | y − 4 | y' = 4 − z | z' = y − 4 | New point |
|---|---|---|---|---|
| A(0,0,0) | −4 | 4 | −4 | A'(0, 4, −4) |
| B(2,3,0) | −1 | 4 | −1 | B'(2, 4, −1) |
| C(5,0,4) | −4 | 0 | −4 | C'(5, 0, −4) |

**Answer (θ = 90°):** A'(0,4,−4), B'(2,4,−1), C'(5,0,−4). The x-coordinates do not change.

## Check yourself

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

1. The line y = 4 (z = 0) is parallel to which axis?
   A. x-axis
   B. y-axis
   C. z-axis
   D. None of them
   Answer: A. x varies freely along the line while y and z are constant, so it is parallel to x. (Key point: Axis direction)

2. The composite transformation is:
   A. T(0,-4,0)·Rx(θ)·T(0,4,0)
   B. T(0,4,0)·Rx(θ)·T(0,-4,0)
   C. Rz(θ) only
   D. T(4,0,0)·Rx(θ)·T(-4,0,0)
   Answer: B. Translate the axis down to the x-axis (y - 4), rotate, then translate back. (Key point: Method)

3. For θ = 90°, A(0,0,0) goes to:
   A. (0, -4, 4)
   B. (0, 0, 0)
   C. (0, 4, -4)
   D. (0, 4, 4)
   Answer: C. y' = 4 - z = 4 and z' = y - 4 = -4. (Key point: Result)

## Related in Computer Graphics

- [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×
- [Rotate line about Y-axis by 90°](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotate-line-y-axis-90) — Computer Graphics, chapter 4
- [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
- [Rotate unit cube 90° about axis AB](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acube-rotate-90-axis-a-b) — 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 rotation matrices (clockwise)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Arotation-matrices-xyz-clockwise) — Computer Graphics, chapter 4

## Related in other subjects

- [Rod with spring released, 90 degree rotation](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Arod-spring-4kg-pivot-b) — Applied Mechanics (Dynamics), chapter 6, asked 2×