# [[Rotation Transformation]] (Mathematics > Linear Algebra)
## Definition
A **rotation transformation** is a linear transformation that rotates vectors in the plane about the origin by a fixed angle. This transformation preserves the length of vectors and the angles between them. It is represented by the rotation matrix:
$
R = \begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}.
$
## Key Concepts
- **Standard Basis:** The standard basis vectors $\{\mathbf{i}, \mathbf{j}\}$ in $\mathbb{R}^2$ are used to illustrate the effect of rotation.
- **Rotation Matrix:** A matrix that rotates any vector by a specified angle $\theta$.
- **Isometry:** Rotation is an isometric transformation, meaning it preserves distances.
- **Orthogonal Transformation:** Rotation matrices are orthogonal with determinant 1.
## Important Properties
1. **Preservation of Lengths:** The transformation does not change the magnitude of vectors.
2. **Angle Preservation:** The angles between vectors remain unchanged after rotation.
3. **Determinant:** The determinant of a rotation matrix is 1, indicating that the area is preserved.
## Essential Formulas
- **Rotation Matrix:**
$
R = \begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}.
$
- **Image of a Vector:**
For any vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$, the rotated vector is:
$
R\mathbf{v} = \begin{pmatrix}
\cos \theta \cdot x - \sin \theta \cdot y \\
\sin \theta \cdot x + \cos \theta \cdot y
\end{pmatrix}.
$
## Core Examples
1. **Basic Example:**
For a rotation by $\theta = 30^\circ$, the rotation matrix is:
$
R = \begin{pmatrix}
\cos 30^\circ & -\sin 30^\circ \\
\sin 30^\circ & \cos 30^\circ
\end{pmatrix}.
$
The images of the standard basis vectors are computed as follows:
$
\mathbf{i}' = R(\mathbf{i}) = \begin{pmatrix}
\cos 30^\circ \\
\sin 30^\circ
\end{pmatrix}, \quad
\mathbf{j}' = R(\mathbf{j}) = \begin{pmatrix}
-\sin 30^\circ \\
\cos 30^\circ
\end{pmatrix}.
$
2. **Advanced Application:**
Visualizing the transformation of the unit square spanned by $\{\mathbf{i}, \mathbf{j}\}$ under $R$, each vertex of the square is rotated by $30^\circ$ about the origin. The square retains its shape and size, demonstrating the isometric property of rotations.
## Related Theorems/Rules
- **Orthogonal Matrices:** Rotation matrices are orthogonal, meaning $R^T R = I$.
- **Isometry Theorem:** Rotations, as isometries, preserve distances and angles in Euclidean space.
## Common Pitfalls
- **Sign Errors:** Mistakenly swapping the sine and cosine values or signs in the rotation matrix.
- **Angle Units:** Confusing degrees with radians when computing trigonometric functions.
- **Order of Operations:** Incorrectly applying the rotation by not correctly multiplying the matrix with the vector.
## Related Topics
- [[Rigid Motions]]
- [[Linear Transformation]]
- [[Standard Matrix of a Linear Transformation]]
- [[Angle_Between_Vectors]]
- [[Determinants]]
- [[Matrix Multiplication]]
- [[Dilation Transformation]]
- **Reflection Transformation:** Another type of linear transformation that flips vectors about a line.
- **Dilation Transformation:** A transformation that scales vectors uniformly.
- **Linear Transformations:** The broader category of functions that include rotations, reflections, and dilations.
## Quick Review Questions
1. What is the effect of applying a rotation matrix to a vector in $\mathbb{R}^2$?
2. How do the images of the standard basis vectors $\mathbf{i}$ and $\mathbf{j}$ under a rotation by $30^\circ$ compare to the original vectors?
***
![[Rotation_Transformation_visualization]]