Search results
Results From The WOW.Com Content Network
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 ...
William Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the ...
Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the remainder of his life to studying and teaching them. From 1844 to 1850 Philosophical Magazine communicated Hamilton's exposition of quaternions. [2] In 1853 he issued Lectures on Quaternions, a comprehensive treatise that also described biquaternions ...
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]
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:
On 4 July 1968 the base was subjected to a heavy People's Army of Vietnam (PAVN) rocket and mortar attack followed by probes on the base perimeter resulting in 5 U.S. and 16 PAVN killed. On 23 February 1969 the base was attacked by PAVN sappers.
When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. A special case of the theorem was first proved by Hamilton in 1853 [6] in terms of inverses of linear functions of quaternions.
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.