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We can express quaternion multiplication in the modern language of vector cross and dot products (which were actually inspired by the quaternions in the first place [14]). When multiplying the vector/imaginary parts, in place of the rules i 2 = j 2 = k 2 = ijk = −1 we have the quaternion multiplication rule:
The field of complex numbers is also isomorphic to three subsets of quaternions.) [22] A quaternion that equals its vector part is called a vector quaternion. The set of quaternions is a 4-dimensional vector space over the real numbers, with { 1 , i , j , k } {\displaystyle \left\{1,\mathbf {i} ,\mathbf {j} ,\mathbf {k} \right\}} as a basis ...
The modified Rodrigues vector is a stereographic projection mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane. The projection of the opposite quaternion − q results in a different modified Rodrigues vector p s than the projection of the original quaternion q .
Vectors and scalars can be added. When a vector is added to a scalar, a completely different entity, a quaternion is created. A vector plus a scalar is always a quaternion even if the scalar is zero. If the scalar added to the vector is zero then the new quaternion produced is called a right quaternion. It has an angle characteristic of 90 degrees.
A direct formula for the conversion from a quaternion to Euler angles in any of the 12 possible sequences exists. [2] For the rest of this section, the formula for the sequence Body 3-2-1 will be shown. If the quaternion is properly normalized, the Euler angles can be obtained from the quaternions via the relations:
which is a quaternion of unit length (or versor) since ‖ ‖ = + + + = Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions =. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with ...
The vector part of a unit quaternion represents the radius of the 2-sphere corresponding to the axis of rotation, and its magnitude is the sine of half the angle of rotation. Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit ...
In mathematics, a versor is a quaternion of norm one (a unit quaternion).Each versor has the form = = + , =, [,], where the r 2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions).