Quaternions were constructed in attempt to extend the idea of rotations in a complex plane to 3D. Each quaternion is a set of four parameters. Why four? Well, any rigid body rotation can be done about a unique axis. So, 3 parameters to specify the axis and 1 for the angle rotated about it. But we have an endless choice of vectors along the axis to represent it. Here comes an applicational constraint from the fact that we want a rotation that doesn’t scale. So we restrict to unit quaternions (quaternions of unit magnitude). In retrospection, this has also simplified the mathematical design for rotation using quaternions. <br \>

So we’ll start with vector notation for rotation and make an attempt to build a mathematical construct for unit quaternions.

In the picture above, x is being rotated to x’ about n in anticlockwise sense. This is equivalent to keeping its component parallel to n preserved and rotating the perpendicular component. Mathematically this is given as,

[[File:Equation13.png|frame|center]]