Inner Product of RoPE-Encoded Token Representations

In the context of positional embeddings, the notation $C(\mathbf{x}, t\theta)$ represents the result of rotating a 2-dimensional vector $\mathbf{x}$ by an angle $t\theta$. If the initial vector is $\mathbf{x} = [4, 2]$ and the rotation angle is $t\theta = \pi/2$, what is the resulting vector?

The notation $C(\mathbf{x}, t\theta)$ is a shorthand for rotating a 2-dimensional vector $\mathbf{x} = [x_1, x_2]$ by an angle $t\theta$. This operation is equivalent to representing the vector as a complex number $(x_1 + i x_2)$ and multiplying it by $e^{it\theta}$. Based on this, which expression correctly represents the first component of the resulting rotated vector?

The transformation represented by the notation $C(\mathbf{x}, t\theta)$, which rotates a 2-dimensional vector $\mathbf{x}$ by an angle $t\theta$, changes the magnitude (or length) of the original vector $\mathbf{x}$.