Description

This app provides a visual demonstration that the multiplication of the complex number $Z = \big|Z\big|\big(cos (\theta) + sin( \theta)i\big)$ by the complex number $W = \big|W\big|\big(cos (\phi) + sin( \phi) i\big)$ can be though of as a combination of scaling of Z by $\big|W\big|$ and then rotating Z counterclockwise through the angle $\phi$.

Instructions

The circles indicate regions of the complex plane where a complex number $Z$ has an absolute value $\big|Z\big| \le 1$ (inside or on the green circle), $\big|Z\big| \le 1.5$ (inside or on the yellow circle) and $\big|Z\big| < 2.25$ (inside or on the magenta circle). To check that the multiplication of $Z$ by $W$ can be thought of as a combination of a scaling and a rotation,

Enter $Z$ by clicking at a point inside the yellow circle.

Enter $W$ by clicking at a second point inside the yellow circle.

An animation will begin that provides a visualization of how the product $\big|ZW\big|$ can be calculated as a combination of a scaling and a rotation.

Details and Options:

• A complex number clicked outside the yellow circle will be normalized so that its absolute value is 1.5.
• Press the Start button after a product has been calculated to clear the generated vectors and curves.
• You can use the Multiply by menu to set the order of the scaling and rotation transformations. You can also change the angle mode (degrees or radians), the thickness of the dynamically generated angle curves, and the animation rate by using the Angle Mode, the Angle Curves and the Animation Rate menus, respectively.

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