View example sentences, synonyms and word forms for Quaternion.
Quaternion meaning
A group or set of four people or things. | A word of four syllables. | A type of four-dimensional hypercomplex number consisting of a real part and three imaginary parts (real multiples of distinct, independent square roots of −1 denoted by i, j and k); commonly used in vector mathematics and as an alternative to matrix algebra in calculating the rotation of three-dimensional objects.
Synonyms of Quaternion
Example sentences (20)
A polar vector can be represented in calculations (for example, when rotated by a quaternion "similarity transform") by a pure imaginary quaternion, with no loss of information, but the two should not be confused.
A quaternion representation of the fermions might be: : A further complication with quaternion representations of fermions is that there are two types of multiplication: left multiplication and right multiplication which must be taken into account.
Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product.
Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos(θ) is a rotation by an angle 2θ, the axis of the rotation being the direction of the imaginary part.
Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to define a vector to mean a pure imaginary quaternion.
In the second equality above, we have identified with a unit quaternion and with a pure quaternion.
Unlike the situation in the complex plane, the conjugation of a quaternion can be expressed entirely with multiplication and addition: : Conjugation can be used to extract the scalar and vector parts of a quaternion.
First then must be put the holy quaternion of the gospels; following them the Acts of the Apostles..
For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable.
In other words, a quaternion squares to −1 if and only if it is a vector (that is, pure imaginary) with norm 1. By definition, the set of all such vectors forms the unit sphere.
Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure).
Multiplying a quaternion by a real number scales its norm by the absolute value of the number.
Notice that replacing i by −i, j by −j, and k by −k sends a vector to its additive inverse, so the additive inverse of a vector is the same as its conjugate as a quaternion.
Perwass also claims here that David Hestenes coined the term "versor", where he is presumably is referring to the GA context (the term versor appears to have been used by Hamilton to refer to an equivalent object of the quaternion algebra).
Quaternions Quaternion Plaque on Broom Bridge The other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843.
See polar decomposition of a quaternion for details of this development of the three-sphere.
The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions.
Then we may state : :Stone–Weierstrass Theorem (quaternion numbers).
The rationale is that combining quaternion transformations is more numerically stable than combining many matrix transformations.
The scalar part of a quaternion is always real, and the vector part is always pure imaginary.