Q56P - Mathematical Methods in Physical Sciences

Question

Describe geometrically the set of points in the complex plane satisfying the following equations.

The angle $\mathit{z}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}$. 

Step-by-Step Solution

Answer

The answer is the positive y-axis.

Step 1: Given Information

The equation is the angle $z=\frac{\pi }{2}$.

Step 2: Definition of the Complex number

A complex number can be represented as:

 
$\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$
   

Where z is the complex number, x and y are real numbers, and i is known as iota, whose value is $\sqrt{\mathbf{-}\mathbf{1}}$.

 

The modulus of a complex number can be calculated as:

 

$\mathbf{\left|}\mathbf{z}\mathbf{\right|}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}$

Step 3: Find the value

The equation is the angle of z,  $\theta =\frac{\pi }{2}$.

 

The complex number is $z=x+yi$.

And 

$\begin{array}{l}z=|z{|}^{i\frac{\mathrm{\pi }}{2}}\\ z=\left|z\right|\left(cos\frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}\right)\\ z=\left|z\right|i\end{array}$ 

 

Hence, in the equation, z is purely imaginary, meaning it is the y-axis, and according to the sign, it will be a positive y-axis.