A rotation by angle $\theta$ about a unit axis

$$\mathbf n=(n_x,n_y,n_z)$$

can be represented by the unit quaternion

$$q=\cos\frac\theta2 +\sin\frac\theta2(n_x\mathbf i+n_y\mathbf j+n_z\mathbf k).$$

Rotating a vector

Represent a vector $\mathbf v=(v_x,v_y,v_z)$ as the pure quaternion

$$p=v_x\mathbf i+v_y\mathbf j+v_z\mathbf k.$$

The rotated vector is represented by

$$p'=qpq^{-1}.$$

Composition

Successive rotations are composed by quaternion multiplication. Because multiplication is noncommutative, order matters just as it does for rotation matrices.

Double cover

The quaternions $q$ and $-q$ represent the same physical rotation. Unit quaternions therefore provide a double cover of the three-dimensional rotation group.

Why quaternions are useful

A unit quaternion uses four components with one normalization constraint, avoids the gimbal-lock singularity of Euler-angle coordinates and supports stable composition without the redundant nine components of a rotation matrix.