What is rotation matrix?

A rotation matrix is a transformation matrix that rotates a vector in a Euclidean space. It is a real square matrix whose determinant is 1 and whose transpose is its inverse. This means that a rotation matrix, denoted by R, satisfies the following conditions:

R is a square matrix.
R is a real matrix (its elements are real numbers).
det(R) = 1 (its determinant is 1).
Rᵀ = R⁻¹ (its transpose is its inverse). Therefore, RᵀR = RRᵀ = I, where I is the identity matrix.

Rotation matrices are used extensively in various fields, including:

Computer graphics: Used to rotate objects in 2D and 3D space.
Robotics: Used to represent the orientation of robots and their joints.
Physics: Used to describe rotations in classical mechanics and quantum mechanics.
Linear Algebra: Provides a matrix representation for rotational transformations.

Properties of Rotation Matrices:

Orthogonal: As mentioned above, Rᵀ = R⁻¹, meaning rotation matrices are orthogonal matrices. This property is crucial for preserving lengths and angles during rotation.
Preserve Lengths and Angles: Applying a rotation matrix to a vector does not change its length or the angles between vectors.
Determinant of 1: The determinant of a rotation matrix is always 1. This distinguishes rotations from reflections, which have a determinant of -1. Matrices with determinant of +1 are called special orthogonal matrices.
Composition of Rotations: Multiplying two rotation matrices results in another rotation matrix. This corresponds to performing two rotations in sequence. The order of multiplication matters, as rotations are generally not commutative.
Inverse of Rotation: The inverse of a rotation matrix (its transpose) represents the inverse rotation – the rotation that undoes the original rotation.

Examples:

2D Rotation Matrix: A 2D rotation matrix that rotates a vector counterclockwise by an angle θ is given by:
```
R = | cos(θ)  -sin(θ) |
    | sin(θ)   cos(θ) |
```

3D Rotation Matrices: In 3D, rotations can be performed around the x, y, or z axes. These are represented by the following matrices:

Rotation around the x-axis by an angle α:

Rx = | 1      0       0     |
     | 0   cos(α)  -sin(α) |
     | 0   sin(α)   cos(α) |

Rotation around the y-axis by an angle β:

Ry = |  cos(β)  0   sin(β) |
     |    0     1      0    |
     | -sin(β)  0   cos(β) |

Rotation around the z-axis by an angle γ:

Rz = | cos(γ)  -sin(γ)  0 |
     | sin(γ)   cos(γ)  0 |
     |    0       0      1 |

Any arbitrary 3D rotation can be represented as a sequence of rotations about these three axes (Euler angles) or using a quaternion representation which avoids gimbal lock, a problem that Euler angles can suffer from.

Understanding rotation matrices is fundamental for working with transformations in multiple dimensions.