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Rotating a vector $\mathbf{x}$ by an angle $\theta$ for $t$ times is equivalent to a single rotation by the angle $t\theta$. This operation is formally expressed as: $$ Ro(\mathbf{x}, t\theta) = \mathbf{x}R_{t\theta} = [\cos t\theta \cdot x_1 - \sin t\theta \cdot x_2 \quad \sin t\theta \cdot x_1 + \cos t\theta \cdot x_2] $$

Formula for Applying RoPE Rotation 't' Times

Given the following scenario from a computer graphics application, determine the final orientation vector of the object.

Rotational Transformation in Animation

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.

Analyzing Rotational Transformation Outcomes

A 2-dimensional vector $\mathbf{x} = [3, 4]$ is transformed by a rotation. The transformation for a vector $[x_1, x_2]$ and a rotation angle $\alpha$ is defined by the formula: 

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

What is the resulting vector if the rotation angle $\alpha$ is $\pi/2$ radians (90 degrees)?

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