Q17P - 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.

${\left(1.7-3.2i\right)}^{\mathbf{2}}$ .

Step-by-Step Solution

Answer

The complex number can be written as: $z=13.13e^{-2.16i}$

 

The graph is mentioned below:

Step 1: Given Information

The complex number is ${\left(1.7-3.2i\right)}^2$.

Step 2: Definition of the Complex number

Real and imaginary numbers combine and form a complex number as:

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$ Here a and b are the real numbers, and z is a complex number

Step 3: Find the values

The complex number is ${\left(1.7-3.2i\right)}^2$ .

Factorize the complex number.

$$\begin{gathered} z={\left(1.7-3.2i\right)}^2 \\ ={\left(1.7\right)}^2-2\times 1.7\times 3.2i+{\left(3.2i\right)}^2 \\ z=2.89-10.24-1.88i \\ z=-7.35-10.88i \end{gathered}$$ 

 

Find x and y:

 $$\begin{gathered} z=-7.35-10.88i \\ x=-7.35 \\ y=-10.88 \end{gathered}$$

 

Find the value of r :

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{(-7.35)}^2+{(-10.88)}^2} \\ =13.13 \end{gathered}$$

 

Find the value of $\theta$ :

 $$\begin{gathered} \theta ={\cos }^{-1}\left(\frac{x}{r}\right) \\ \theta ={\cos }^{-1}\left(-\frac{7.35}{13.13}\right) \\ =-124° \\ =-2.16 \mathrm{rad} \\ \mathrm{z}=13.13{\mathrm{e}}^{-2.16\mathrm{i}} \end{gathered}$$

 

The complex number can be written as: $\mathrm{z}=13.13{\mathrm{e}}^{-2.16\mathrm{i}}$

 

The graph is mentioned below: