Analysis of a Vector Transformation Property

Consider the transformation applied to a 2-dimensional vector $\mathbf{x} = [x_1, x_2]$ by an angle $\theta$, defined by the formula:

$\mathrm{Result} = [\cos(\theta) \cdot x_1 - \sin(\theta) \cdot x_2, \sin(\theta) \cdot x_1 + \cos(\theta) \cdot x_2]$

This transformation alters the magnitude (length) of the vector $\mathbf{x}$.