Chapter 14.5, Problem 21WE

a.

To determine

Sketcha diagram for the given vectors.

Explanation of Solution

Given:

The given vectors are

  $z=1+i$ and $z^2$ .

Calculation:

To plot complex numbers in the complex plane. In the complex plane , the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

The given vector $z^2={\left(1+i\right)}^2=2i$ .

The sketchof the vectors $z=1+i$ and $z^2$ is given below.

  

In the graph green dot represents the vector $z=1+i$ and red dot represents the vector $z^2$ .

b.

To determine

Sketch a diagram for the given vectors.

Explanation of Solution

Given:

The given vectors are

  $z=1+i$ and $z^3$ .

Calculation:

To plot complex numbers in the complex plane. In the complex plane , the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

The given vector $z^3={\left(1+i\right)}^3=-2+2i$ .

The sketch of the vectors $z=1+i$ and $z^3$ is given below .

  

In the graph green dot represents the vector $z=1+i$ and blue dot represents the vector $z^3$ .

c.

To determine

Sketch a diagram for the given vectors.

Explanation of Solution

Given:

The given vectors are

  $z=1+i$ and $z^4$ .

Calculation:

To plot complex numbers in the complex plane. In the complex plane , the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

The given vector $z^4={\left(1+i\right)}^4=-4$ .

The sketch of the vectors $z=1+i$ and $z^4$ is given below .

  

In the graph green dot represents the vector $z=1+i$ and red dot represents the vector $z^4$ .

d.

To determine

Sketch a diagram for the given vectors.

Explanation of Solution

Given:

The given vectors are

  $z=1+i$ and $z^5$ .

Calculation:

To plot complex numbers in the complex plane. In the complex plane , the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

The given vector $z^5={\left(1+i\right)}^5=-4-4i$ .

The sketch of the vectors $z=1+i$ and $z^5$ is given below .

  

In the graph green dot represents the vector $z=1+i$ and red dot represents the vector $z^5$ .

e.

To determine

Sketch a diagram for the given vectors.

Explanation of Solution

Given:

The given vectors are

  $z=1+i$ and $z^6$ .

Calculation:

To plot complex numbers in the complex plane. In the complex plane , the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

The given vector $z^6={\left(1+i\right)}^2=-8i$ .

The sketch of the vectors $z=1+i$ and $z^6$ is given below .

  

In the graph green dot represents the vector $z=1+i$ and red dot represents the vector $z^6$ .

f.

To determine

Sketch a diagram for the given vectors.

Explanation of Solution

Given:

The given vectors are

  $z=1+i$ and $z^7$ .

Calculation:

To plot complex numbers in the complex plane. In the complex plane , the horizontal axis is called the real axis, and the vertical axis is called the imaginary axis.

The given vector $z^7={\left(1+i\right)}^7=8-8i$ .

The sketch of the vectors $z=1+i$ and $z^7$ is given below .

  

In the graph green dot represents the vector $z=1+i$ and red dot represents the vector $z^7$ .