Stitz-Zeager_College_Algebra_e-book

We get x r sin r y r cos r which can be written

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Unformatted text preview: ep process. The multiplication of |z | by |w| can be interpreted as magnifying12 the distance |z | from z to 0, by the factor |w|. Adding the argument of w to the argument of z can be interpreted geometrically as a rotation of β radians counter-clockwise.13 Focusing on z and w from 12 13 Assuming |w| > 1. Assuming β > 0. 11.7 Polar Form of Complex Numbers 851 Example 11.7.3, we can arrive at the product zw by plotting z , doubling its distance from 0 (since π |w| = 2), and rotating 23 radians counter-clockwise. The sequence of diagrams below attempt to geometrically describe this process. Imaginary Axis Imaginary Axis 6i 6i 5i π 6 zw = 8cis 4i π 6 z |w| = 8cis + 5i 2π 3 4i 3i z |w| = 8cis π 6 3i 2i 2i π 6 z = 4cis i i 01 2 3 4 5 6 7 Real Axis −7 −6 −5 −4 −3 −2 −1 Multiplying z by |w| = 2. 01 2 3 4 5 Rotating counter-clockwise by Arg(w) = Visualizing zw for z = 4cis π 6 2π 3 and w = 2cis Real Axis 6 7 2π 3 radians. . z z We may also visualize division similarly. Here, the formula w = ||w|| cis(α − β ) may be interpreted...
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