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The conjugate of a product of two quaternions is the product of the conjugates in the reverse order. That is, if p and q are quaternions, then (pq) ∗ = q ∗ p ∗, not p ∗ q ∗. The conjugation of a quaternion, in stark contrast to the complex setting, can be expressed with multiplication and addition of quaternions:
Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation. Multiplication of a scalar and a vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did ...
The product of two quaternionic matrices A and B also follows the usual definition for matrix multiplication. For it to be defined, the number of columns of A must equal the number of rows of B . Then the entry in the i th row and j th column of the product is the dot product of the i th row of the first matrix with the j th column of the ...
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]
Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of i (red), j (green) and k (blue). In the SVG file, hover over or click a path to highlight it. The next step in the construction is to generalize the multiplication and conjugation operations. Form ordered pairs (a, b) of complex numbers a and b, with multiplication ...
He then carved the basic rules for multiplication into the bridge: i 2 = j 2 = k 2 = ijk = −1. 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]
Since unit quaternions are subject to two algebraic constraints, unit quaternions are standard to represent rigid transformations. [2] Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length.
The red cycle also reflects that i 2 = e, i 3 = i and i 4 = e. In group theory, the quaternion group Q 8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {,,,,,} of the quaternions under multiplication.
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