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The positive and negative basis vectors form the eight-element quaternion group. Graphical representation of products of quaternion units as 90° rotations in the planes of 4-dimensional space spanned by two of {1, i, j, k}. The left factor can be viewed as being rotated by the right factor to arrive at the product. Visually i ⋅ j = − (j ⋅ i)
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]
These two scalars (negative and positive unity) can be thought of as scalar quaternions. These two scalars are special limiting cases, corresponding to versors with angles of either zero or π. These two scalars are special limiting cases, corresponding to versors with angles of either zero or π.
Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.
The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group. The study of integral quaternions began with Rudolf Lipschitz in 1886, whose system was later simplified by Leonard Eugene Dickson; but the modern system was published by Adolf Hurwitz in 1919.
Like rotation matrices, quaternions must sometimes be renormalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of renormalizing a quaternion, however, is much less than for normalizing a 3 × 3 matrix. Quaternions also capture the spinorial character of rotations in three dimensions.
This can be seen by comparing the basis to the quaternion basis, or from the above product which is identical to the quaternion product, except for a change of sign which relates to the negative products in the bivector scalar product A · B. Other quaternion properties can be similarly related to or derived from geometric algebra.
The axes of the original frame are denoted as x, y, z and the axes of the rotated frame as X, Y, Z.The geometrical definition (sometimes referred to as static) begins by defining the line of nodes (N) as the intersection of the planes xy and XY (it can also be defined as the common perpendicular to the axes z and Z and then written as the vector product N = z × Z).