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Derivation of 2D Vector Rotation using Complex Numbers

The rotation of a 2D vector x=(x1,x2)\mathbf{x} = (x_1, x_2) by an angle tθt\theta can be derived using complex number multiplication. First, the vector is represented as a complex number x′=x1+ix2x' = x_1 + i x_2. The rotation is then performed by multiplying x′x' by eitθ=cos⁡tθ+isin⁡tθe^{it\theta} = \cos t\theta + i \sin t\theta. The expansion of this product reveals the components of the new rotated vector:

xRtθ↦(x1+ix2)eitθ=(x1+ix2)(cos⁡tθ+isin⁡tθ)=(cos⁡tθ⋅x1−sin⁡tθ⋅x2)+i(sin⁡tθ⋅x1+cos⁡tθ⋅x2)\begin{aligned} \mathbf{x}R_{t\theta} & \mapsto (x_1 + i x_2) e^{it\theta} \\ &= (x_1 + i x_2)(\cos t\theta + i \sin t\theta) \\ &= (\cos t\theta \cdot x_1 - \sin t\theta \cdot x_2) + i(\sin t\theta \cdot x_1 + \cos t\theta \cdot x_2) \end{aligned}

The real part of the result corresponds to the new x-coordinate, and the imaginary part corresponds to the new y-coordinate of the rotated vector.

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Updated 2026-04-29

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