Solved Problems on Rotation Matrix
Example 1: If A (1, -2) is rotated in the counterclockwise direction by 60°, what are the coordinate values?
Solution:
We know that, [Tex]\begin{bmatrix} x’\\ y’ \end{bmatrix}=\begin{bmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}[/Tex]
Thus, [Tex]\begin{bmatrix} x’\\ y’ \end{bmatrix}=\begin{bmatrix} cos60 & -sin60 \\ sin60 & cos60 \end{bmatrix} \begin{bmatrix} 1\\ -2 \end{bmatrix}[/Tex]
On solving we get, (x’, y’) = (1/2 + √3, 1 + √3/2)
Example 2: If B (2, 3) is rotated in the clockwise direction by 90°, what are the coordinate values?
Solution:
We know that [Tex]\begin{bmatrix} x’\\ y’ \end{bmatrix}=\begin{bmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}[/Tex]
Thus, [Tex]\begin{bmatrix} x’\\ y’ \end{bmatrix}=\begin{bmatrix} cos(-90) & -sin(-90) \\ sin(-90) & cos(-90) \end{bmatrix} \begin{bmatrix} 2\\ 3 \end{bmatrix}[/Tex]
[Tex]\begin{bmatrix} x’\\ y’ \end{bmatrix}=\begin{bmatrix} 0& 1\\ -1& 0 \end{bmatrix} \begin{bmatrix} 2\\ 3 \end{bmatrix}[/Tex]
On solving we get, (x’, y’) = (3, -2)
Example 3: If C (5, 2, 6) is rotated in the counterclockwise direction by 180° about the x-axis, what are the coordinate values?
Solution:
Since, [Tex]\begin{bmatrix} x’\\y’\\z’ \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & cos(\gamma) & -sin(\gamma) \\ 0 & sin(\gamma) & cos(\gamma) \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix}[/Tex]
Therefore, [Tex]\begin{bmatrix} x’\\y’\\z’ \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & cos(180) & -sin(180) \\ 0 & sin(180) & cos(180) \end{bmatrix} \begin{bmatrix} 5\\2\\6 \end{bmatrix}[/Tex]
[Tex]\begin{bmatrix} x’\\y’\\z’ \end{bmatrix}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 &0 & -1 \end{bmatrix} \begin{bmatrix} 5\\2\\6 \end{bmatrix}[/Tex]
On solving we get (x’, y’, z’) = (5, -2, -6)
Rotation Matrix
Rotation Matrix is a matrix used to perform a rotation in a coordinate space. Rotation matrices are widely used in various fields, including computer graphics, robotics, physics, and navigation systems, to describe and manipulate the orientation of objects in space. In this article, we will discuss Rotation Matrices including 2D and 3D rotation matrices.
Table of Content
- What is a Rotation Matrix?
- Definition of Rotation Matrix
- Example of Rotation using Matrix
- Representation of Rotations in Mathematics
- 2D Rotation Matrix
- Derivation of the 2D Rotation Matrix
- Rotating Points in a 2D Plane
- 3D Rotation Matrix
- Derivation of the 3D Rotation Matrix
- Rotating Points in 3D Coordinates
- Properties of Rotation Matrices
- Euler’s Rotation Theorem
- Applications of Rotation Matrices
- Conclusion