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:

$\cos(÷{φ}{2})+i⋅a_x⋅\sin(÷{φ}{2})+j⋅a_y⋅\sin(÷{φ}{2})+k⋅a_z⋅\sin(÷{φ}{2})$

where the axis is normalized so that .

where the quaternion is normalised so that $(\cos(÷{φ}{2}))²+a_x²⋅(\sin(÷{φ}{2}))²+a_y²⋅(\sin(÷{φ}{2}))²+a_z²⋅(\sin(÷{φ}{2}))²=1$ .

let's say that is: .

Then follows by taking the square root:

$q_w=\cos(÷{φ}{2})$

$q_x=a_x⋅\sin(÷{φ}{2})$

$q_y=a_y⋅\sin(÷{φ}{2})$

$q_z=a_z⋅\sin(÷{φ}{2})$

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:

$P=q⋅p⋅\conj{q}$

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