There are countless articles on the Web and elsewhere which describe their application to rotations in three dimensional space. Here we briefly mention the application to 'ordinary' four dimensional space and the relation between Hamilton quaternions and the Pauli quanternions of four dimensional space time.
In four dimensions (and higher dimensions), a general rotation is defined by a rotation plane and an angle: all vector components normal to the plane are unchanged; the two components in the plane are rotated by an angle which may be defined by two vectors with a common origin in that plane. To calculate effect of this rotation on an arbitary vector, we first need a formula for a square root of a quaternion:
Square root of a quaternion and other functions
For quaternion Q
Define the functions:
Then a square root of quaternion Q is:
Four Dimensional rotations
Let Qa and Qb be quaternions with a common origin; to rotate Qc in the plane defined by Qa and Qb thru the angle spanned by Qa and Qb:
define
Then to rotate any quaternion Qc by this angle:
Miscellaneous geometry
Area of parallelogram defined by Qa and Qb
Volume of parallelpiped
A quaternion perpendicular to three quaternions Qa,Qb Qc and having length equal to the volume of the parallelpiped defined by Qa,Qb,Qc is (Sudbery)
Relation with Pauli quaternions
The Hamilton multiplication rules differ from the Pauli matrix rules only by a factor of i. It is possible to formulate special relativity with Hamilton quaternions having complex coefficients(called biquaternions) and indeed it was first done that way(Silberstein). It turns out that the formulae of general relativity are simpler with the Pauli quaternions. There is also a very interesting (and possibly significant) relation between the Pauli quaternions and three dimensional Clifford Algebra.