A 2D vector can be rotated by representing it as a complex number and multiplying it by another complex number that represents the rotation. Arrange the following steps in the correct logical order to derive the coordinates of a vector (x₁, x₂) after it has been rotated by an angle θ.
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Ch.2 Generative Models - Foundations of Large Language Models
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Definition of C(x, tθ) Notation for Rotated Tokens
Component-wise Vector Rotation in Complex Space
A 2D vector (x, y) can be represented by the complex number z = x + iy. To rotate this vector by an angle θ, it is multiplied by the complex number e^(iθ) = cos(θ) + i sin(θ). If the vector (3, 4) is rotated by 90 degrees (π/2 radians) using this method, what are the coordinates of the new, rotated vector?
A 2D vector can be rotated by representing it as a complex number and multiplying it by another complex number that represents the rotation. Arrange the following steps in the correct logical order to derive the coordinates of a vector
(x₁, x₂)after it has been rotated by an angleθ.Mapping Complex Multiplication to Rotated Vector Components