Chapter 8, Problem 30RE

a.

To determine

Graph the complex number in the complex plane.

  $-10i$

Answer to Problem 30RE

The graph of complex number in the complex plane $-10i$ is,

  

Explanation of Solution

Given:

  $-10i$

Concept Used:

Desmos Graph.

According to the given details,

  $z=-10i$

  

Conclusion:

Hence, graph is

  

b.

To determine

The modulus and argument.

  $-10i$

Answer to Problem 30RE

The modulus and argument is $r=10,\theta =\frac{\pi }{2}$

Explanation of Solution

Given:

  $-10i$

Concept Used:

Modulus and argument: $r=\sqrt{x^2+y^2}$

Polar form: $z=r\left(\cos \theta +i\sin \theta \right)$

Calculation:

The given complex number is $z=-10i$ ,

So, write it in the polar form,

  $z=r\left(\cos \theta +i\sin \theta \right)$

Now evaluate $r$ and $\theta$ , as know that the formula for these terms,

  $r=\sqrt{x^2+y^2}$

So, from the equation,

  $\begin{array}{l}x=0\\ y=-10\end{array}$

Substituting these values in the above formula,

  $\begin{array}{l}r=\sqrt{{\left(0\right)}^2+{\left(-10\right)}^2}\\ r=\sqrt{0+100}\\ r=10\end{array}$

Now evaluate, $\theta$

  $\begin{array}{l}\theta =\arctan \left(-10\right)\\ \theta =\frac{\pi }{2}\end{array}$

Conclusion:

Hence, the area is $\frac{1}{2}\left(a_1{b}_3-a_3{b}_1-a_2{b}_3+a_3{b}_2+a_1{b}_2+a_2{b}_1\right)$ .

c.

To determine

The polar form of $-10i$ .

Answer to Problem 30RE

The polar form of $-10i$ is $z=10\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$

Explanation of Solution

Given:

  $-10i$

Concept Used:

Polar form: $z=r\left(\cos \theta +i\sin \theta \right)$

Calculation:

  $z=-10i$

In polar form,

  $\begin{array}{l}z=r\left(\cos \theta +i\sin \theta \right)\\ r=10\\ \theta =\frac{\pi }{2}\end{array}$

So now write these calculated values in standard equation of polar form,

  $z=10\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$

Conclusion:

Hence, the polar form of $-10i$ is $z=10\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$ .