A way of respresenting the orientation of a 3 dimension body in space, using a 4D object
A 4-dimensional extension of the complex numbers
The quaternion describes the orientation of the object’s local reference frame relative to the global reference frame
A harder to visualize and more abstract, but easier to use alternative to Euler Angles; not suspectible to the bugs and edge cases of euler angles, e.g. Gimbal Lock.
Used heavily in (3D) Computer Graphics

Definition

is precisely the used in complex numbers. and function the same as , but are along different axes.

Operations

Let , correspond to a vector in . This is the vector that we want to rotate.

Let , be the quaternion that we will use to manipulate

Left multiplying by rotates, and translates negative to the axis of rotation
Right multiplying by rotates the same amount, and translates positive to the axis of rotation
Therefore results in that rotation, twice, with the translation cancelled out. This is the form of quaternion rotation.

Let

The following is a quaternion that rotates about by degrees