Search results
Results from the WOW.Com Content Network
For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x. The orthogonal group, consisting of all proper and improper rotations, is generated by reflections.
The set of all orthogonal matrices of size n with determinant +1 is a representation of a group known as the special orthogonal group SO(n), one example of which is the rotation group SO(3). The set of all orthogonal matrices of size n with determinant +1 or −1 is a representation of the (general) orthogonal group O(n).
The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1.
Orthogonal matrices are important for a number of reasons, both theoretical and practical. The n × n orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by O(n), which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold.The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices.
An example of an r = 3 parameter group is the matrix group SO(3) consisting of all 3 × 3 orthogonal matrices with unit determinant. A possible parametrization of this group is in terms of Euler angles: x = ( α , β , γ ) {\displaystyle \mathbf {x} =(\alpha ,\beta ,\gamma )} (see e.g., this article for the explicit form of an element of SO(3 ...