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A common method for conceptualizing vector rotation involves mapping vectors from a two-dimensional Euclidean space ($$R^2$$) to the complex plane ($$C$$). This is achieved through a bijective linear transformation, which allows a 2D vector $$\mathbf{x} = [x_1, x_2]$$ to be represented as a single complex number.

Representing 2D Vector Rotation in Complex Space

The rotation of a 2D vector by an angle $$t\theta$$ can be represented as multiplication in the complex plane. This is done by multiplying the vector's complex number representation by the term $$e^{it\theta}$$. The identity $$e^{it\theta} = \cos t\theta + i \sin t\theta$$, known as Euler's formula, provides the crucial link to re-express this operation in terms of the vector's components and trigonometric functions.

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 vector `x = [5, -2]`?

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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