Google

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 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.

Analyzing a Flawed Vector Rotation

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?

Learn Before

Related