Mapping Complex Multiplication to Rotated Vector Components
When a 2D vector represented by the complex number z = x₁ + i x₂ is rotated by an angle θ, the operation is performed by multiplying z by e^(iθ) = cos θ + i sin θ. The result of this multiplication is the complex number (x₁ cos θ - x₂ sin θ) + i(x₁ sin θ + x₂ cos θ). Explain how the real and imaginary parts of this resulting complex number correspond to the coordinates of the new, rotated vector.
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Ch.2 Generative Models - Foundations of Large Language Models
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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