Quaternions can be represented in several ways. One of the ways is similar to the way complex
numbers are represented:
q ≡ q4 + q1i + q2j + q3k,
in which q1 , q2 , q3 and q4 , are real numbers, and i, j, and k, are unit “vectors” which obey similar rules to the vectors of the same names found in vector analysis, but with an additional similarity to the i of complex arithmetic which equals − 1 . The multiplication rules for i , j , and k are depicted
conceptually as follows:
That is, i j = + k, j k = + i, etc. , from figure 1(a) , and j i = - k, i k = -j , etc., from figure 1(b) . Expressed in this form, the multiplication rules are very easy to remember. Note that the cross products of i, j , and k obey the rules of vector cross product multiplication, where, for example, given the orthogonal axes, x, y, and z: x × y = z, y × z = x , and z × x = y .
Quaternions are also useful for rotating vectors around an axis without gimbal lock and for performing rotations between coordinate frames. This latter characteristics with coordinate frames makes quaternions the ideal tool for determining the attitude of objects in animations and in real life, such as the attitude of an orbiting satellite. The attachments to this page include a detailed pdf document on the use of quaternions in aerospace.