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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:
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 product of a quaternion with its conjugate is its common norm. [63] The operation of taking the common norm of a quaternion is represented with the letter N. By definition the common norm is the product of a quaternion with its conjugate. It can be proven [64] [65] that common norm is equal to the square of the tensor of a quaternion ...
By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation.
Quaternion variable theory differs in some respects from complex variable theory. For example: The complex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, non-analytic operation. Indeed, conjugation changes the orientation of plane figures, something that arithmetic functions do not ...
Addition is defined pairwise. The product of two pairs of quaternions (a, b) and (c, d) is defined by (,) (,) = (, +) , where z* denotes the conjugate of the quaternion z. This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs
Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were r*q rather than qr*, the formula for multiplication of an element by its conjugate would not yield a real number.
which is a quaternion of unit length (or versor) since ‖ ‖ = + + + = Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions =. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with ...