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The tangent space to S 2 at a point m is naturally identified with the vector subspace of R 3 consisting of all vectors orthogonal to m. It follows that a vector field Y on S 2 can be seen as a map Y : S 2 → R 3 , which satisfies Y ( m ) , m = 0 , ∀ m ∈ S 2 . {\bigl \langle }Y(m),m{\bigr \rangle }=0,\qquad \forall m\in \mathbf {S} ^{2}.
Quaternions give a simple way to encode this [7] axis–angle representation using four real numbers, and can be used to apply (calculate) the corresponding rotation to a position vector (x,y,z), representing a point relative to the origin in R 3. Euclidean vectors such as (2, 3, 4) or (a x, a y, a z) can be rewritten as 2 i + 3 j + 4 k or a x ...
A quaternion of the form a + 0 i + 0 j + 0 k, where a is a real number, is called scalar, and a quaternion of the form 0 + b i + c j + d k, where b, c, and d are real numbers, and at least one of b, c, or d is nonzero, is called a vector quaternion. If a + b i + c j + d k is any quaternion, then a is called its scalar part and b i + c j + d k ...
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
The perifocal coordinate system (with unit vectors p, q, w), against the reference coordinate system (with unit vectors I, J, K) The perifocal coordinate (PQW) system is a frame of reference for an orbit. The frame is centered at the focus of the orbit, i.e. the celestial body about which the orbit is centered.
The quaternion product AC is a linear transformation by the operator A of the components of the quaternion C, therefore there is a matrix representation of A operating on the vector formed from the components of C. Assemble the components of the quaternion C = c 0 + C into the array C = (C 1, C 2, C 3, c 0). Notice that the components of the ...
By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so (3) to its Lie group SO(3) .
In mathematics, a versor is a quaternion of norm one (a unit quaternion).Each versor has the form = = + , =, [,], where the r 2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions).