3D Rotation Matrices

In 3D, an orientation is a 3 by 3 rotation matrix whose columns are the body's x, y, and z axes written in the world frame. Unlike 2D, the order you apply rotations in changes where you end up.

Rotation about x Rotation about y Rotation about z Order (about the fixed world axes) Show the other order as a faint copy

Drag the 3D view to look from another angle.

The basic rotations

$$R_x(\alpha) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & -\sin\alpha \\ 0 & \sin\alpha & \cos\alpha \end{bmatrix} \quad R_y(\beta) = \begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix} \quad R_z(\gamma) = \begin{bmatrix} \cos\gamma & -\sin\gamma & 0 \\ \sin\gamma & \cos\gamma & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Rotating first about x, then y, then z, all about the fixed world axes, gives $R = R_z(\gamma)\,R_y(\beta)\,R_x(\alpha)$. The first rotation sits on the right.

What to notice

Start with 90° about x and 90° about z, then switch the order. The same two rotations put the body in a completely different pose, shown by the faint copy. Matrix multiplication does not commute in 3D, and neither do rotations.

Every rotation matrix is orthonormal: its columns have length 1 and are at right angles to each other, and its inverse is its transpose, $R^{-1} = R^\top$. The readout shows each column as one of the body's axes.