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Determining an Unknown Rotation Angle
A 4-dimensional real vector undergoes a component-wise rotation where its components are grouped into consecutive pairs, with each pair treated as a complex number (a + ib) and rotated independently. The initial vector is x = (4, 3, 5, 12). The first pair of components is rotated by an angle of π/2 radians. The second pair is rotated by an unknown angle, θ. The final, transformed vector is y = (-3, 4, -5, -12). Based on this information, what is the value of the unknown rotation angle θ?
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Ch.3 Prompting - Foundations of Large Language Models
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Analysis in Bloom's Taxonomy
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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.