Quaternions are like complex numbers.

The number of dimensions in a quaternion number is 4.

Multiplication over quaternions is NOT commutative.

Multiplication over quaternions is associative.

Representation for example:

Where i is the complex root, j is another basis root and k is yet another basis root.

## Rotations

If we have an axis angle representation consisting of an angle and an axis represented by a vector (a_x,a_y,a_z) of unit length, the equivilant quaternion representation is:

where the axis is normalized so that .

where the quaternion is normalised so that .

let's say that is: .

Then follows by taking the square root:

## Apply rotation to a point

To apply rotation to a point, represent the point as i, j and k factors (the real part is 0), then the new point P is, given the old point p and a rotation quaternion q:

... where conj is the conjugate operator, flipping the signs of all the non-real components of the quaternion.