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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

The rotation of a 2D vector $\mathbf{x} = (x_1, x_2)$ by an angle $t\theta$ can be derived using complex number multiplication. First, the vector is represented as a complex number $x' = x_1 + i x_2$. The rotation is then performed by multiplying $x'$ by $e^{it\theta} = \cos t\theta + i \sin t\theta$. The expansion of this product reveals the components of the new rotated vector:
$$
\begin{aligned}
\mathbf{x}R_{t\theta} & \mapsto (x_1 + i x_2) e^{it\theta} \\
&= (x_1 + i x_2)(\cos t\theta + i \sin t\theta) \\
&= (\cos t\theta \cdot x_1 - \sin t\theta \cdot x_2) + i(\sin t\theta \cdot x_1 + \cos t\theta \cdot x_2)
\end{aligned}
$$
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.

Derivation of 2D Vector Rotation using Complex Numbers

A 2D vector `(x, y)` can be represented as a complex number `x + iy`. Rotating this vector counter-clockwise by an angle `θ` is equivalent to multiplying the complex number by `cos(θ) + i sin(θ)`. Given the vector `(3, 4)`, what is its new position after a 90-degree counter-clockwise rotation?

A 2D vector is represented in the complex plane by the number `z = x + iy`. If this complex number `z` is multiplied by the imaginary unit `i`, what are the coordinates of the resulting new vector? Based on this result, describe the specific geometric transformation that multiplication by `i` performs on the original vector.

Geometric Effect of Multiplication by the Imaginary Unit

A programmer is attempting to rotate a 2D vector, represented as the complex number `z = x + iy`, by an angle `θ`. They implement this by multiplying `z` by the complex number `r = cos(θ) + i`. Their results are incorrect. Identify the error in the complex number `r` used for rotation and explain the unintended geometric transformation that results from this operation.

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