Vectors can rotate around {rotation operator} own axis, coordinate axis, or any axis. Vectors can rotate around points.
rotation only
Complex rotation operator is 1 - (i/2) * (Ax * dt(y, z) + Ay * dt(z, x) + Az * dt(x, y)), where t is time and Ax = |0, 1, 1, 0|, Ay = |0, -1, i, 0|, and Az = |1, 0, 0, -1|. A^2 = 1. Ax * Ay = - Ay * Ax = i * Az. Operator changes vector direction but not length or origin {active transformation}.
Product of two rotation operators gives result of two rotations. Rotations have inverses. Rotation and inverse rotation result in no rotation, so product is zero.
translation
Complex translation operators can changes reference frame, but vector maintains direction and length {passive transformation}. cos(t/2) - (i * sin(t/2)) * (Ax * cos(a) + Ay * cos(b) + Az * cos(c)), where t is rotation angle, Ax = |0, 1, 1, 0|, Ay = |0, -1, i, 0|, and Az = |1, 0, 0, -1|. A^2 = 1. Ax * Ay = - Ay * Ax = i * Az. i^2 = j^2 = k^2 = i * j * k = 1. j * k = 1. k * j = -1. k * i = j. i * k = -j. i * j = k. j * i = - k.
reflection
If rotation angle is t, rotation is equivalent to two successive reflections in two planes that meet at angle t/2.
Mathematical Sciences>Calculus>Vector>Operations
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224