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The real quaternion 1 is the identity element. The real quaternions commute with all other quaternions, that is aq = qa for every quaternion q and every real quaternion a. In algebraic terminology this is to say that the field of real quaternions are the center of this quaternion algebra.
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
The quaternion group has the unusual property of being Hamiltonian: Q 8 is non-abelian, but every subgroup is normal. [4] Every Hamiltonian group contains a copy of Q 8. [5] The quaternion group Q 8 and the dihedral group D 4 are the two smallest examples of a nilpotent non-abelian group.
The quaternions are a non-commutative extension of the complex numbers which have numerous applications in mathematics, physics, and computer graphics The main article for this category is Quaternion .
non-zero quaternions with multiplication N 0 0 H: 4 S 3 = Sp(1) quaternions of absolute value 1 with multiplication; topologically a 3-sphere: Y 0 0 isomorphic to SU(2) and to Spin(3); double cover of SO(3) Im(H) 3 GL(n,R) general linear group: invertible n×n real matrices: N Z 2 – M(n,R) n 2: GL + (n,R) n×n real matrices with positive ...
The quaternion elements vary continuously over the unit sphere in ℝ 4, (denoted by S 3) as the orientation changes, avoiding discontinuous jumps (inherent to three-dimensional parameterizations) Expression of the rotation matrix in terms of quaternion parameters involves no trigonometric functions
Each quaternion has a tensor, which is a measure of its magnitude (in the same way as the length of a vector is a measure of a vectors' magnitude). When a quaternion is defined as the quotient of two vectors, its tensor is the ratio of the lengths of these vectors.
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).