Chapter 1: Q10P (page 1)

First simplify each of the following numbers to the $x + i y$ form or to the $r e^{i θ}$ form. Then plot the number in the complex plane.
$\frac{3 i - 7}{i + 4}$ .

Short Answer

Expert verified

The complex number becomes $\sqrt{\frac{58}{17}} e^{142.8 i}$ .

Step by step solution

Given Information

The complex number is $\frac{3 i - 7}{i + 4}$ .

Definition of the Complex number

A complex number is a number that is made up of both real and imaginary numbers.

Find the value

The complex number is $\frac{3 i - 7}{i + 4}$ .

Rationalize the complex number.

$z = \frac{- 7 + 3 i}{4 + i} \times \frac{4 - i}{4 - i} = \frac{- 28 + 19 i - 3 i^{2}}{16 - i^{2}} = \frac{- 25 + 19 i}{17} z = - \frac{25}{17} + \frac{19 i}{17}$

Find x and y.

$z = x + i y z = - \frac{25}{17} + \frac{19 i}{17} x = - \frac{25}{17} y = \frac{19}{17}$

Find the value of r.

localid="1664175531640" $r = \sqrt{x^{2} + y^{2}} r = \sqrt{\frac{625}{289} + \frac{361}{289}} = \sqrt{\frac{986}{289}} = \sqrt{\frac{58}{17}}$

Find the value of $θ$ .

$θ = \cos^{- 1} (\frac{x}{r}) θ = \cos^{- 1} (- \frac{- \frac{25}{17}}{\sqrt{\frac{58}{17}}}) θ = 142 . 8^{0}$

The complex number becomes $\sqrt{\frac{58}{17}} e^{142.8 i}$ .

The graph is mentioned below.

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First simplify each of the following numbers to the $x + i y$ form or to the $r e^{i θ}$ form. Then plot the number in the complex plane.
$\frac{3 i - 7}{i + 4}$ .

Short Answer

Step by step solution

Given Information

Definition of the Complex number

Find the value

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Most popular questions from this chapter

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First simplify each of the following numbers to the x+iy form or to the reiθ form. Then plot the number in the complex plane. 3i-7i+4 .

Short Answer

Step by step solution

Given Information

Definition of the Complex number

Find the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Fields in Physics

Translational Dynamics

Solid State Physics

Electromagnetism

Torque and Rotational Motion

Study anywhere. Anytime. Across all devices.

First simplify each of the following numbers to the $x + i y$ form or to the $r e^{i θ}$ form. Then plot the number in the complex plane.
$\frac{3 i - 7}{i + 4}$ .