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Chapter 6: Problem 43

Find the product of the complex numbers. Leave answers in polar form. $$\begin{aligned} &z_{1}=1+i\\\ &z_{2}=-1+i \end{aligned}$$

Short Answer

Expert verified
The product of the two complex numbers in polar form is \( z = \sqrt{2} e^{i(\frac{\pi}{4} - \frac{\pi}{4})} \), which can be simplified to \( z = \sqrt{2}\) .

Step by step solution

01

Convert Complex Numbers From Cartesian Form to Polar Form

Complex number \( z_1 = 1 + i \) can be represented as \( r_1 e^{i\theta_1} \) in polar form. Whereas, \( r_1 = \sqrt{Re^2 + Im^2} \), and \( \theta_1 = arctan(\frac{Im}{Re}) = arctan(1) \). Similarly, Complex number \( z_2 = -1 + i \) can be represented as \( r_2 e^{i\theta_2} \) in polar form. Where \( r_2 = \sqrt{Re^2 + Im^2}\), and \( \theta_2 = arctan(\frac{Im}{Re}) = arctan(-1) \).
02

Multiply The Complex Numbers

Once both numbers are in polar form, we can multiply them by multiplying the magnitudes and adding the angles. That is, \( z = z_1 \times z_2 = r_1 \times r_2 e^{i(\theta_1 + \theta_2)} \) .
03

Write The Answer in Polar Form

Now, the product of two complex numbers is obtained, present your final answer in polar form. Your final answer should look like \( a + bi\), where \( a = r cos \theta \) and \( b = r sin \theta \) .

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Most popular questions from this chapter

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}-\mathbf{w})$$

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j},$$ to prove the given property. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

A force is given by the vector \(\mathbf{F}=5 \mathbf{i}+7 \mathbf{j} .\) The force moves an object along a straight line from the point (8,11) to the point \((18,20) .\) Find the work done if the distance is measured in meters and the force is measured in newtons.

Write an equation in point-slope form and general form for the line passing through (-2,5) and perpendicular to the line whose equation is \(x-4 y+8=0\) (Section \(1.5,\) Example 2 )

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$5 \mathbf{u} \cdot(3 \mathbf{v}-4 \mathbf{w})$$
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