Search results
Results from the WOW.Com Content Network
12 Quaternion algebras ... which have dimension 1, 2, and 4 ... of all quaternions is a vector space over the real numbers with dimension 4. [c] Multiplication of ...
p ↦ q p for q = 1 + i + j + k / 2 on the unit 3-sphere. Note this one-sided (namely, left) multiplication yields a 60° rotation of quaternions. The length of is √ 3, the half angle is π / 3 (60°) with cosine 1 / 2 , (cos 60° = 0.5) and sine √ 3 / 2 , (sin 60° ≈ 0.866). We are therefore dealing with a ...
1.4.2 Right versor. ... [12] [13] [14] A tensor can be ... multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion ...
Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors when computing the product of matrices. The identity for this multiplication is, as expected, the diagonal matrix I = diag(1, 1, ... , 1). Multiplication follows the usual laws of associativity and distributivity. The trace of a matrix ...
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F [1] [2] that has dimension 4 over F.Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
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).
In the diagrams for D 4, the group elements are marked with their action on a letter F in the defining representation R 2. The same cannot be done for Q 8, since it has no faithful representation in R 2 or R 3. D 4 can be realized as a subset of the split-quaternions in the same way that Q 8 can be viewed as a subset of the quaternions.
Assemble the components of the quaternion C = c 0 + C into the array C = (C 1, C 2, C 3, c 0). Notice that the components of the vector part of the quaternion are listed first and the scalar is listed last. This is an arbitrary choice, but once this convention is selected we must abide by it.