Comparison of Complex and Euclidean Representations of RoPE Rotation

Consider a 4-dimensional real vector (\mathbf{x} = (1, 0, 0, 1)). A transformation is applied that groups the vector's components into consecutive pairs, treats each pair as a complex number (a + ib), and rotates them independently. The first pair (components 1 and 2) is rotated by an angle of (\pi/2) radians, and the second pair (components 3 and 4) is rotated by an angle of (\pi) radians. What is the resulting vector after this transformation?

Determining an Unknown Rotation Angle

A transformation is applied to a d-dimensional real vector by grouping its components into d/2 consecutive pairs. Each pair is treated as representing a point in a 2D plane and is independently rotated by a specific, non-zero angle. True or False: This transformation will always change the Euclidean norm (magnitude) of the original d-dimensional vector.