Q7P - Mathematical Methods in Physical Sciences

Question

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

$\left(\frac{3+i}{2+i}\right)$

Step-by-Step Solution

Answer

The complex number can be written as $\sqrt{2}{e}^{-i0.141}$.

 

The graph is mentioned below:

 

Step 1: Given Information

Thecomplex number is $\left(\frac{3+i}{2+i}\right)$.

Step 2: Definition of the Complex number.

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

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

 

Step 3: Find the value

Factorize the complex number.

 $$\begin{gathered} z=\frac{3+i}{2+i}\times \frac{2-i}{2-i} \\ =\frac{6-i-i^2}{4-i^2} \\ =\frac{7}{5}-\frac{i}{5} \end{gathered}$$

 

Find x and y

 $$\begin{gathered} x=\frac{7}{5} \\ =-\frac{1}{5} \end{gathered}$$

Find the value of r .

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{\frac{49}{25}+\frac{1}{25}} \\ =\sqrt{\frac{50}{25}} \\ =\sqrt{2} \end{gathered}$$ 

 

Find the value of  .

 $$\begin{gathered} \theta ={\sin }^{-1}\left(\frac{y}{r}\right) \\ \theta ={\sin }^{-1}\left(\frac{{\displaystyle \frac{1}{5}}}{\sqrt{2}}\right) \\ =-0.141 \\ \left(\frac{3+i}{2+i}\right)=\sqrt{2}{e}^{-i0.141} \end{gathered}$$

 

The complex number can be written as $\sqrt{2}{e}^{-i0.141}$.

 

The graph is mentioned below as: