An AI developer is implementing a positional embedding layer. They need to apply a 2D rotation to vector components, but it is critical that the transformation does not change the vector's length (magnitude), as this would distort the information encoded in the embedding. They are considering two potential transformation functions for a vector $x = \begin{bmatrix} x_1 & x_2 \end{bmatrix}$ and an angle $\theta$:

• Function A: $y = \begin{bmatrix} \cos \theta \cdot x_1 - \sin \theta \cdot x_2 & \sin \theta \cdot x_1 + \cos \theta \cdot x_2 \end{bmatrix}$
• Function B: $y = \begin{bmatrix} \cos \theta \cdot x_1 + \sin \theta \cdot x_2 & \sin \theta \cdot x_1 + \cos \theta \cdot x_2 \end{bmatrix}$

Analyze both functions. Which function should the developer choose to ensure the vector's magnitude is preserved after the transformation? Justify your answer by explaining why the chosen function works and the other one fails.