The Quaternion Calculator includes functions associated with quaternion mathematics. These include the following:*QUATERNION ROTATIONS*

- Quaternion Addition
- Quaternion Subtraction
- Quaternion Multiplication
- Quaternion Magnitude
- Quaternion Versor
- Quaternion Conjugate
- Quaternion Inverse
- Quaternion of Rotation
- Vector Rotation

## Quaternions

Quaternions can be represented in several ways. One of the ways is similar to the way complex

numbers are represented:

q ≡ q_{4} + q_{1}**i** + q_{2}**j** + q_{3}**k**,

in which q_{1} , q_{2} , q_{3} and q_{4} , 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 .

**Note:** Quaternions are not commutative, and the following should be noted:

**q1*q2 ≠ q2 * q1 q1*q2 ≠ -q2 * q1**, but

(q1 * q2) * q3 = q1 * (q2 * q3)

(q1 * q2) * q3 = q1 * (q2 * q3)