Chapter 14.4, Problem 10OE

To determine

To find: What happens to the absolute value and the amplitude of a complex number when it is multiplied by given number.

Answer to Problem 10OE

Absolute value of the product remains same while amplitude increases by ${180}^0$ . .

Explanation of Solution

Given information:

A complex number is multiplied by $-1$ .

Concepts used: Multiplication and division of two complex numbers.

If $w=a\left(\cos \alpha +i\sin \alpha \right)\text{ and }z=b\left(\cos \beta +i\sin \beta \right)$ then −

  $\begin{array}{l}wz=ab\left[\cos \left(\alpha +\beta \right)+i\sin \left(\alpha +\beta \right)\right]\\ \frac{w}{z}=\frac{a}{b}\left[\cos \left(\alpha -\beta \right)+i\sin \left(\alpha -\beta \right)\right]\end{array}$

Calculation:

Let us suppose a complex number is given as $w=r\left(\cos \theta +i\sin \theta \right)$ .

This is multiplied by $z=-1$

  $\Rightarrow z=1\left(\cos {180}^0+i\sin {180}^0\right)$

{Converting into polar form}

  $\begin{array}{l}w=r\left(\cos \theta +i\sin \theta \right)\text{ and }z=1\left(\cos {180}^0+i\sin {180}^0\right)\\ \text{So, }wz=r\cdot 1\left[\cos \left(\theta +{180}^0\right)+i\sin \left(\theta +{180}^0\right)\right]\\ \Rightarrow wz=r\left[\cos \left(\theta +{180}^0\right)+i\sin \left(\theta +{180}^0\right)\right]\end{array}$

Hence, absolute value of the product remains same while amplitude increases by ${180}^0$ .