A 2D vector $\mathbf{x} = [x_1, x_2]$ is transformed using the formula:

New Vector = $[\cos(\alpha) \cdot x_1 - \sin(\alpha) \cdot x_2, \quad \sin(\alpha) \cdot x_1 + \cos(\alpha) \cdot x_2]$.

For what value of the angle $\alpha$ (in radians, within the interval $[0, 2\pi]$) will the new vector be equal to $[ -x_1, -x_2 ]$? Justify your answer by explaining how this specific value for $\alpha$ affects the sin and cos terms in the formula to produce the desired outcome.