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Rotation as Complex Number Multiplication
The rotation of a 2D vector by an angle can be represented as multiplication in the complex plane. This is done by multiplying the vector's complex number representation by the term . The identity , known as Euler's formula, provides the crucial link to re-express this operation in terms of the vector's components and trigonometric functions.
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Ch.2 Generative Models - Foundations of Large Language Models
Foundations of Large Language Models
Foundations of Large Language Models Course
Computing Sciences
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Rotation as Complex Number Multiplication
A two-dimensional vector, such as
v = [a, b], can be represented as a single number in the complex plane. This is done by mapping the first component of the vector to the real part of the complex number and the second component to the imaginary part. Given this transformation, which of the following complex numbers correctly represents the vectorx = [5, -2]?When mapping two-dimensional vectors to the complex plane, it is possible for two distinct vectors to be represented by the exact same complex number.
A common method for analyzing vector rotation involves representing two-dimensional vectors as complex numbers. Match each 2D vector on the left with its corresponding representation in the complex plane on the right.
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Derivation of 2D Vector Rotation using Complex Numbers
A 2D vector
(x, y)can be represented as a complex numberx + iy. Rotating this vector counter-clockwise by an angleθis equivalent to multiplying the complex number bycos(θ) + i sin(θ). Given the vector(3, 4), what is its new position after a 90-degree counter-clockwise rotation?Geometric Effect of Multiplication by the Imaginary Unit
Analyzing a Flawed Vector Rotation