Composition Property of Rotations

Representing 2D Vector Rotation in Complex Space

Definition of the 2D Rotation Matrix

Consider a system where the position of a token in a sequence is encoded by rotating its initial vector embedding, x. The total angle of rotation is directly proportional to the token's position, m. If the vector for a token at position 3 is obtained by rotating x by a total angle of 3θ, what is the correct transformation to find the vector for the same token at position 9?

A positional encoding system represents a token's position by sequentially rotating its initial vector embedding, denoted as x, by a fixed angle θ for each step forward in a sequence. Arrange the following vector states to show the correct order of transformations for a token as its position advances from 1 to 3.

In a system that encodes sequential position, an initial vector x is transformed to represent position m by applying a cumulative rotation, resulting in vector v_m. Similarly, the vector for position m+1 is v_{m+1}. Based on this mechanism, what is the direct geometric transformation that relates v_m to v_{m+1}?