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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 9: Q3E (page 437)

Question: Write $z = - 1 + i$ in polar form as $z = r e^{i 胃}$ .

Short Answer

Expert verified

Answer

The solution is $z = \sqrt{2} e^{\frac{3 蟺}{4}}$ .

Step by step solution

Given

$z = - 1 + i$

Explanation of the solution

Consider the equation as follows.

$z = - 1 + i$

So, the following condition holds as follows.

$z = ({(1)}^{2} + {(- 1)}^{2}) e^{t a n^{- 1} (\frac{1}{- 1}) i} z = (1 + 1) e^{t a n^{- 1} (- i)} z = 2 e^{t a n^{- 1} (- i)} z = \sqrt{2} e^{\frac{3 蟺}{4}}$

Thus, $z = - 1 + i$ can be written in the polar form as $z = \sqrt{2} e^{\frac{3 蟺}{4}}$ .

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Most popular questions from this chapter

Use theorem 9.3.13 to solve the initial value problem

$\frac{d \overset{鈫�}{x}}{d t} = (\begin{matrix} 2 & 3 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}) \overset{鈫�}{x}$ with $\overset{鈫�}{x} (0) = [\begin{matrix} 2 \\ 1 \\ - 1 \end{matrix}]$ .

Hint: Find first $x_{3} (t)$ , then $x_{2} (t)$ , and finally $x_{1} (t)$ .

Question: Consider the interaction of two species in a habitat. We are told that the change of the populations $\overset{鈫�}{x} (t) a n d \overset{鈫�}{y} (t)$ can be moderated by the equation

$| \begin{matrix} \frac{d x}{d t} = 1.4 x - 1.2 y \\ \frac{d x}{d t} = 1.8 x - 1.4 y \end{matrix} |$

where timeis a measured in years.

What kind of intersection do we observe (symbiosis, competition, or predator-prey)?
Sketch the phase portrait for the system. From the nature of the problem, we are interested only in the first quadrant.
What will happen in the long term? Does the outcome depend on the initial populations? If so, how?

Solve the nonlinear differential equations in Exercises 6through 11 using the method of separation of variables:Write the differential equation $\frac{d x}{d t} = f x$ as $\frac{d x}{f x} = d t$ and integrate both sides.

7. $\frac{d x}{d t} = x^{2}, x (0) = 1$

Describe the behavior of your solution as t increases.

Solve the differential equation $f " (t) + 2 f' (t) + f (t) = 0$ and find all the real solutions of the differential equation.

Let be an matrix anda scalar. Consider the following two systems:

$\frac{d \overset{鈫�}{x}}{dt} = A \overset{鈫�}{x} (l) \frac{d \overset{鈫�}{c}}{dt} = (A + {kl}_{n}) \overset{鈫�}{c} (ll)$

Show that if $\overset{鈫�}{x} (t)$ is a solution of the system (l)then role="math" localid="1659701582223" $\overset{鈫�}{c} (t) = e^{k t} \overset{鈫�}{x} (t)$ is a solution of the system (ll).

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Question: Write z=-1+i in polar form as z=rei胃.

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Question: Write $z = - 1 + i$ in polar form as $z = r e^{i 胃}$ .