Q11P - Mathematical Methods in Physical Sciences

Question

Question:First simplify each of the following numbers to the x+iy form or to the $\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}$ form. Then plot the number in the complex plane.

17-12i .

Step-by-Step Solution

Answer

The complex number becomes $z=20.8e^{-e0.615i}$ .

 

The graph is mentioned below as:

 

Step 1: Given Information

Thecomplex number is 17 - 12i.

Step 2: Definition of the Complex number

A complex number is a number that consists of real and imaginary numbers and is written in the form as:

$z=a+ib$ where a and b are real, and z is the complex number.

Step 3: Find the values

The complex number is 17 - 12i .

Find x and y:

x = 17

y = -12 

Find the value of r.

 $$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{\left(17\right)}^2+{\left(-12\right)}^2} \\ =20.8 \end{gathered}$$

 

Find the value of  $\theta$:

 $$\begin{gathered} \theta ={\cos }^{-1}\left(\frac{x}{r}\right) \\ \theta ={\cos }^{-1}\left(\frac{17}{20.8}\right) \\ =-35.18° \\ =-.0615 \mathrm{rad} \\ \mathrm{z}=20.8{\mathrm{e}}^{-.0615\mathrm{i}} \end{gathered}$$

 

The complex number becomes $\mathrm{z}=20.8{\mathrm{e}}^{-.0615\mathrm{i}}$ .

 

The graph is mentioned below: