Permuted Input Vector in RoPE

A transformation is applied to a d-dimensional vector $\mathbf{x}$ at position $t$ using a set of frequencies $\theta$. The transformation is defined by the following element-wise formula:

Analyzing Components of a Vector Transformation

Consider the element-wise formula for applying a rotational transformation to a d-dimensional vector $\mathbf{x}$. The formula involves summing two vectors derived from element-wise products.

True or False: The first vector in the sum is the product of the original input vector $\mathbf{x}$ and a sine vector, and the second vector is the product of a permuted input vector (e.g., $[-x_2, x_1, \dots]$) and a cosine vector.