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For two elements a 1 + b 1 i + c 1 j + d 1 k and a 2 + b 2 i + c 2 j + d 2 k, their product, called the Hamilton product (a 1 + b 1 i + c 1 j + d 1 k) (a 2 + b 2 i + c 2 j + d 2 k), is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of ...
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.
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 ...
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 mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. [1] Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebraic constraints are used ...
1.4.2 Right versor. 1.4.3 ... multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion and a vector or of two ...
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b).
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.