3D Rotation Matrices — Rotations Around X, Y, Z Axes

3D Rotation Matrices

In 3D, there are three basic rotation matrices, one for each coordinate axis.

Rotation Around X-Axis

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

Rotation Around Y-Axis

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

Rotation Around Z-Axis

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

Composing Rotations

A combined rotation is the product of individual rotations. The order matters because matrix multiplication is not commutative:

R = Rz(c) * Ry(b) * Rx(a)  (ZYX Euler angles)

Properties

• All 3D rotation matrices have determinant = 1
• They are orthogonal: R^T = R^(-1)
• They form the SO(3) group (Special Orthogonal group in 3D)
• Composing rotations gives another rotation

Gimbal Lock

When using Euler angles (three sequential axis rotations), certain angle combinations cause two axes to align, losing one degree of freedom. This is called gimbal lock. Quaternions are often used to avoid this problem while still representing rotations compactly.